Volume 39, Issue 10, October 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models
View Description Hide DescriptionWe study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard.

Closure of field operators, asymptotic Abelianness, and vacuum structure in hyperfunction quantum field theory
View Description Hide DescriptionThough, in contrast to standard quantum field theory(QFT), the tensoralgebra over the testfunction space of hyperfunction quantum field theory has no local structure, the localization properties of states on this algebra can be used to derive asymptotic Abelianness in spacelike directions. Again, in contrast to standard QFT, the closure of (Hermitian) field operators can destroy localization properties. This problem is addressed in a natural modification of the definition of the closure, called the local closure. This allows one, in conjunction with asymptotic Abelianness, to define a proper reduction of the field algebra to the subspace of the translation invariant states, and to investigate the dimension of this subspace.

Improving the convergence and estimating the accuracy of summation approximants of expansions for Coulombic systems
View Description Hide DescriptionThe convergence of largeorder expansions in where is the dimensionality of coordinate space, for energies of Coulomb systems is strongly affected by singularities at and Padé–Borel approximants with modifications that completely remove the singularities at and remove the dominant singularity at are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function where are the the expansion coefficients of the energy with singularity structure removed at farther from the origin and thereby accelerate summation convergence. The groundstate energies of He and are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using is considered. The standard deviation of results calculated with different values of the shift parameter is proposed as a measure of summation accuracy.

Fermion realization of the nuclear model
View Description Hide DescriptionA fermion realization of the nuclear model, which complements the traditional bosonic representation, is developed. A recursive process is presented in which symplectic matrix elements of arbitrary onebody fermion operators between states of excitation and in the same or in different symplectic bands are related back to valence shell matrix elements, which can be evaluated by standard shell model techniques. Matrix elements so determined may be used to calculate observables such as electron scatteringform factors which carry detailed structural information on nuclear wave functions.

Operator formalism for bosonic beta–gamma fields on general algebraic curves
View Description Hide DescriptionAn operator formalism for bosonic β–γ system on arbitrary algebraic curves is introduced. The classical degrees of freedom are identified and their commutation relations are postulated. The explicit realization of the algebra formed by the fields is given in the Hilbert space equipped with a bilinear form. The construction is based on the “Gaussian” representation for β–γ system on the complex sphere [AlvarezGaumé et al., Nucl. Phys. B 311, 333 (1988)]. Detailed computations are provided for two and fourpoint correlation functions.

Application of the discrete Wentzel–Kramers–Brillouin method to spin tunneling
View Description Hide DescriptionA discrete version of the Wentzel–Kramers–Brillouin (WKB) method is developed and applied to calculate the tunnel splittings between classically degenerate states of spin Hamiltonians. The results for particular model problems are in complete accord with those previously found using instanton methods. The discrete WKB method is more elementary and also yields wave functions.

Quantum coordinates of an event in local quantum physics
View Description Hide DescriptionRecently Toller [quantph/9702060 (1997)] has proposed, using the formalism of positiveoperatorvalued measures, a possible definition of quantum coordinates for events in the context of quantum mechanics. In this short note we analyze this definition from the point of view of local algebras in the framework of local quantum theories.

Kontsevich integral for Vassiliev invariants from Chern–Simons perturbation theory in the lightcone gauge
View Description Hide DescriptionWe analyze the structure of the perturbative series expansion of Chern–Simons gauge theory in the lightcone gauge. After introducing a regularization prescription that entails the consideration of framed knots, we present the general form of the vacuum expectation value of a Wilson loop. The resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges. It also contains the Kontsevich integral for Vassiliev invariants of framed knots.

A Lie algebra for closed strings, spin chains, and gauge theories
View Description Hide DescriptionWe consider quantum dynamical systems whose degrees of freedom are described by matrices, in the planar limit Examples are gauge theories and the M(atrix)theory of strings. States invariant under are “closed strings,” modeled by traces of products of matrices. We have discovered that the invariant operators acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinitedimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCDmodels) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model,models satisfying the Dolan–Grady condition and the chiral Potts model. Thus our cyclix Lie algebra describes the dynamical symmetries of quantum spin chain systems, largegauge field theories, and the M(atrix)theory of strings.

Operator formulation of Wigner’s Rmatrix theories for the Schrödinger and Dirac equations
View Description Hide DescriptionThe Rmatrix theories for the Schrödinger and Dirac equations are formulated in the language of integral operators. In the nonrelativistic theory the central role is played by an integral operator relating function values to normal derivatives on a surface S of a closed volume V, inside which the function satisfies the Schrödinger equation at energy E. In the relativistic theory, the same role is played by two integral operators, and , linking on the surface S values of upper and lower components of spinor wave functions satisfying in the volume V the Dirac equation at energy E. Systematic procedures for constructing the operators and generalizing the methods due to Kapur and Peierls and to Wigner, are presented.

Extension of Bertrand’s theorem and factorization of the radial Schrödinger equation
View Description Hide DescriptionThe Bertrand’s theorem is extended, i.e., closed orbits still may exist for central potentials other than the power law Coulomb potential and isotropic harmonic oscillator. It is shown that for the combined potential when (and only when) is the Coulomb potential or isotropic harmonic oscillator, closed orbits still exist for suitable angular momentum. The correspondence between the closeness of classical orbits and the existence of raising and lowering operators derived from the factorization of the radial Schrödinger equation is investigated.

A constructive index theorem, phase operators, and phasedifference operators
View Description Hide DescriptionA constructive reversed index theorem concerning the polar decomposition of an operator into a product of a unitary exponential phase operator and a Hermitian amplitude operator is investigated. Its applications to the construction of Hermitian phase operators of fermions and phasedifference operators between bosons or fermions are presented. Specifically, a Hermitian operator is constructed for the phase difference between a singlemode fermion and boson, which is justified by the fact that it is exactly the interaction part of the Hamiltonian of the JaynesCummings Model. Furthermore, a Hermitian phase operator of a singlemode boson can also be defined referring to a singlemode fermion. All those quantized phases and phasedifferences are found to obey a quantum addition rule instead of the ordinary commutative addition rule.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Integrable and solvable manybody problems in the plane via complexification
View Description Hide DescriptionA simple prescription allows us to transform, by appropriate complexification, any onedimensionalmanybody problem (describing “motions on a line”) with Newtonian (“acceleration equals forces”) equations of motion featuring forces which depend analytically on the positions, and possibly on the velocities, of the particles, and which are moreover scaling or translationinvariant, into a twodimensionalmanybody problem (describing “motions in the plane”) with rotationinvariantequations of motion. If the original (onedimensional) model is Hamiltonian, the twodimensional model is also Hamiltonian (in fact, biHamiltonian). In this manner, starting from known integrable or solvable onedimensional manybody problems, integrable or solvable manybody problemsin the plane, generally featuring much richer dynamics, are obtained. Several examples are exhibited. Finally another, more direct, prescription is outlined, to transform by complexification almost any onedimensional manybody problem into a rotationinvariantmanybody problemin the plane.

Two families of nonstandard Poisson structures for Newton equations
View Description Hide DescriptionTwo families of nonstandard twodimensional Poisson structures for systems of Newton equations are studied. They are closely related either with separable systems or with the socalled quasiLagrangian systems. A theorem characterizing the general form of biHamiltonian formulation for separable systems in two and in n dimensions is formulated and proved.

Stochastic construction of a Feynman path integral representation for Green’s functions in radiative transfer
View Description Hide DescriptionA construction of a stochastic, Lagrangian path integral representation is presented for classical Green’s functions or propagators for radiative transfer in random scattering medias. As Fermat rays comprising the initial collimated pulse (e.g., from a laser) enter the medium, they evolve into a bundle of random paths or trajectories due to scattering. Stochastically, this can be interpreted as a bundle of randomly evolving Markov trajectories traced out by a gas or ensemble of “Browniantype” classical point photons undergoing multiple scatterings. The path integral is structurally of the same form as a Wick rotated Euclidean quantum Feynman integral with direct optical analogs of the Hamiltonian and Lagrangian emerging. However, the optical stochastic integral is real and is defined in real time becoming exponentially damped rather than oscillatory. The calculation also constitutes an alternative mathematical derivation of the time dependent diffusion equation from the radiative transfer theory. The limits of the path integral representation give a maximally diffused Gaussian distribution of the heat kernal form (the large scatter limit) and a Beer’s law exponential decay corresponding to the extremal Fermat rays obeying Euler–Lagrange equations (the zero scatter limit). The approach also highlights the direct structural analogs between classical radiative transfer and optical diffusion in real time (t) and ordinary quantum mechanics in Euclidean time
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


The monotony criterion for a finite size scaling analysis of phase transitions
View Description Hide DescriptionWe propose a new criterion to analyze the order of phase transitions within a finite size scaling analysis. It refers to response functions like order parameter susceptibilities and the specific heat and states different monotony behavior in volume for first and secondorder transitions close to the transition point. The criterion applies to analytical and numerical studies of phase diagrams including tricritical behavior.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Continuous iteration of dynamical maps
View Description Hide DescriptionA precise meaning is given to the notion of continuous iteration of a mapping. Usual discrete iterations are extended into a dynamical flow which is a homotopy of them all. The continuous iterate reveals that a dynamic map is formed by independent component modes evolving without interference with each other. An application to turbulent flow suggests that the velocity field assumes nonseparable values.

Vertex operator representation of the soliton tau functions in the Toda models by dressing transformations
View Description Hide DescriptionWe study the relation between the groupalgebraic approach and the dressing symmetry one to the soliton solutions of the Toda field theory in dimensions. Originally, solitons in the affine Toda models were found by Olive, Turok, and Underwood. Single solitons are created by exponentials of elements which addiagonalize the principal Heisenberg subalgebra. Alternatively, Babelon and Bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sineGordon model. In this paper we show the equivalence between these two methods to construct solitons in the Toda models.

Higher dimensional Painlevé integrable models from the Kadomtsev–Petviashvili equation
View Description Hide DescriptionAfter embedding the Kadomtsev–Petviashvili equation in higher dimensions and extending the Painlevé analysis approach to a new form such that the coefficients of the expansion around the singular manifold possess conformal invariance and contain explicit new space variables, we can get infinitely many Painlevé integrable models in dimensions and higher dimensions. Some concrete higher dimensional modified Korteweg–de Vries type of extensions are given. Whether the models are Lax integrable or integrable under other meanings remain still open.

Rational solutions of the KP hierarchy and the dynamics of their poles. II. Construction of the degenerate polynomial solutions
View Description Hide DescriptionA general approach to constructing the polynomial solutions satisfying various reductions of the Kadomtsev–Petviashvili (KP) hierarchy is described. Within this approach, the reductions of the KP hierarchy are equivalent to certain differential equations imposed on the τfunction of the hierarchy. In particular, the lreduction and the kconstraint as well as their generalized counterparts are considered. A general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the KdV and Gardner equations, the Boussinesq and classical Boussinesq systems, the NLS and Yajima–Oikawa equations. It is shown that the degenerate rational solutions of the KP hierarchy are related to stationary manifolds of the Calogero–Moser (CM) hierarchy of dynamical systems. The scattering dynamics of interacting particles in the CM systems may become complicated due to an anomalously slow fractionalpower rate of the particle motion along the stationary manifolds.
