Volume 40, Issue 6, June 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Moyal–Nahm equations
View Description Hide DescriptionVarious aspects of the Nahm equations in three and seven dimensions are investigated. The residues of the variables at simple poles in the sevendimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large N limit of these equations is taken by replacing the commutators by Moyal brackets, and a set of nontrivial solutions in a generalized form of Wigner distribution functions is obtained.

A rigorous path integral for quantum spin using flatspace Wiener regularization
View Description Hide DescriptionAdapting ideas of Daubechies and Klauder [J. Math. Phys. 26, 2239 (1985)] we derive a rigorous continuum pathintegral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space of Lebesgue squareintegrable functions on the Euclidean plane The pathintegral formula emerges from the standard Feynman–Kac–Itô formula for the Schrödinger semigroup in the ultradiffusive limit of the underlying Brownian bridge on In a similar vein, a pathintegral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.

On the geometric equivalence of certain discrete integrable Heisenberg ferromagnetic spin chains
View Description Hide DescriptionUsing a discrete curve mapping procedure, we reformulate the problem of nonlinear spin dynamics in three different discrete Heisenberg ferromagneticspin chain models with different magnetic interactions. The procedure helps to rewrite the Landau–Lifshitz equations that govern the dynamics of spins in these ferromagnetic spin systems as equivalent to the integrable discretization of the completely integrable nonlinear Schrödinger family of equations. The elementary spin excitations in these spin systems are governed by lattice solitons.

The geometry of coherent states
View Description Hide DescriptionWe examine the geometry of the state space of a relativistic quantum field. The mathematical tools used involve complex algebraic geometry and Hilbert spacetheory. We consider the Kähler geometry of the state space of any quantum field theory based on a linear classical fieldequation. The state space is viewed as an infinitedimensional complex projective space. In the case of boson fields, a special role is played by the coherent states, the totality of which constitutes a nonlinear submanifold C of the projective Fock space We derive the metric on C induced from the ambient Fubini–Study metric on Arguments from differential geometric, algebraic, and Kählerian points of view are presented, leading to the result that the induced metric is flat, and that the intrinsic geometry of C is Euclidean. The coordinates for the singleparticle Hilbert space of solutions are shown to be complex Euclidean coordinates for C. A transversal intersection property of complex projective lines in with C is derived, and it is shown that the intrinsic geodesic distance between any two coherent states is strictly greater than the corresponding geodesic distance in the ambient Fubini–Study geometry. The functional metric norm of a difference field is shown to give the intrinsic geodesic distance between two coherent states, and the metric overlap expression is shown to measure the angle subtended by two coherent states at the vacuum, which acts as a preferred origin in the Euclidean geometry of C. Using the flatness of C we demonstrate the relationship between the manifold complex structure on and the quantum complex structure viewed as an active transformation on the singleparticle Hilbert space. These properties of C hold independently of the specific details of the singleparticle Hilbert space. We show how C arises as the affine part of its compactification obtained by setting the vacuum part of the state vector to zero. We discuss the relationship between unitary orbits and geodesics on C and on We show that for a Fock space in which the expectation of the total number operator is bounded above, the coherent state submanifold is Kähler and has finite conformal curvature.

A new type of loop independence and quantum Yang–Mills theory in two dimensions
View Description Hide DescriptionThe expectation values of Wilson loop products for the pure Euclidean Yang–Mills theory on given by Ashtekar et al. in the article “ Quantum Yang–Mills Theory in Two Dimensions: A Complete Solution” [J. Math. Phys. 38, 5453 (1997)] are determined directly for all piecewise analytic loops. For that purpose we enlarge their calculations from quadratic lattices to general floating lattices introducing a new kind of loop independence and slightly modifying the regularization scheme.

Quantization of massive vector fields in curved space–time
View Description Hide DescriptionWe develop a cannonical quantization for massive vector fields on a globally hyperbolic Lorentzian manifold.

Exact spectral values for discrete quantum pendulumintegrals
View Description Hide DescriptionFor specific choice of parameters the spectrum of the discrete quantum pendulumintegral contains the eigenvalues of a finite matrix which depends analytically on the flux. Under natural continuity assumptions these eigenvalues include the spectral values which may be obtained by the algebraic Bethe ansatz.

Exact ground state of several body problems with an body potential
View Description Hide DescriptionThe exact (bosonic) ground state and a class of excited states are obtained for several Calogerotype Nbody problems in D dimensions when the N bodies are also interacting via an Nbody potential of the form

Eigenvalue problems of Ginzburg–Landau operator in bounded domains
View Description Hide DescriptionIn this paper we study the eigenvalue problems for the Ginzburg–Landau operator with a large parameter in bounded domains in under gauge invariant boundary conditions. The estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed. These results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity.

Slater sum for central field problems characterized by its swave component alone
View Description Hide DescriptionFor the hydrogenlike atom, with central potential partial differential equations exist for the Slater sum and for its swave component It is shown that Z can be eliminated, to lead to a result in which is solely characterized by A similar situation is exhibited for the threedimensional isotropic harmonic oscillator, for which closed forms of both and can be obtained explicitly. Finally, a third central field problem is considered in which independent electrons are confined within a sphere of radius R, but are otherwise free. We are able to derive explicitly for this model the swave component The full Slater sum then is also analyzed in some detail.

Flavor symmetry of the tensor Dirac theory
View Description Hide DescriptionRecently, the Dirac and Einstein equations were unified in a tetrad formulation of a Kaluza–Klein model with gauge group In this model, the selfadjoint modes of the tetrad describe gravity, whereas the isometric modes of the tetrad together with a scalar field describe fermions. This model gives precisely the usual Dirac–Einstein Lagrangian. In this paper we generalize the tensorDiractheory to the larger gauge group acting on bispinors. We show that each subgroup of corresponds to a different factorization of the secondorder Klein–Gordon equation into a firstorder Dirac equation. Since the Noether currents are different for each factorization, the solutions describe different flavors of fermions. We show that electric charge,lepton number, and baryon number are conserved in this generalization of the Diractheory.

Superconnections and the Higgs field
View Description Hide DescriptionWithin the mathematical framework of Quillen, one interprets the Higgs field as part of the superconnection D on a superbundle. We propose to take as superbundle the exterior algebra obtained from a Hermitian vector bundle of rank n with structure group and study the curvature The Euclidean action, at most quadratic in F and invariant under gauge transformations, depends on central charges. Spontaneous symmetry breaking is related to a nonvanishing constant scalar curvature in the ground state, where is the Higgs condensate. The Higgs model is nothing but the familiar Ginzburg–Landau theory, whereas the Higgs model relates to the electroweak theory (without matter fields). The present formulation leads to the relation for the coupling constants, the formula for the Weinberg mixing angle, and the ratio for the masses of and the Higgs boson. Experimentally observed deviations are attributed to loop corrections.

Momentum and spin of a particle with spin unity
View Description Hide DescriptionThe first part of this paper deals with the problem of expressing a given boost as the product of an operator which leaves the momentum of a given particle, massive or massless, invariant and a helicity conserving operator. This is done by multiplying boosts and rotations only. In this way more information is obtained than by applying the operators to the relevant vectors. An operator with the property of conserving spin while perhaps changing momentum is developed. The second part of the paper defines a spin angular momentum operator leaving two fourvectors invariant. These vectors are interpreted as the momentum and spin vectors of a massive particle and the theory is extended to accommodate a massless particle. This theory emphasizes the close association of spin and momentum in a relativistic theory. The spin angular momentum operator of a massless particle has invariance properties which resemble some of those of the gauge invariant electromagnetictheory.

Zero curvature formalism of the fourdimensional Yang–Mills theory in superspace
View Description Hide DescriptionThe supersymmetric descent equations in superspace are discussed by means of the introduction of two operators which allow to decompose the supersymmetric covariant derivatives as BRS commutators.

Complete sets of Bloch and Wannier functions composed of oscillator eigenfunctions
View Description Hide DescriptionWe discuss Bloch and Wannier functions related to oscillator eigenfunctions. In particular, we construct complete sets of mutually orthogonal Bloch and Wannier functions. We show that they can be expressed in several ways in terms of theta functions and their derivatives. We also analyze their localization properties and discuss expectation values for specifically chosen Hamiltonians.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Symmetries of discrete dynamical systems involving two species
View Description Hide DescriptionThe Lie point symmetries of a coupled system of two nonlinear differentialdifference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to ten dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.

Dissipative canonical flows in classical and quantum mechanics
View Description Hide DescriptionA theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the overall canonical/symplectic structure on the algebra is preserved. We study both classical and quantum systems and the importance of physical interpretation in the Stratonovich interpretation is stressed. We find the natural formulation of quantum dissipative systems to be given in terms of quantum stochastic calculus. This treatment allows for a physically meaningful treatment of both constant and nonlinear dissipation. As an application, we quantize a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator.

A new first integral for a binary rigid body collision of arbitrarily short duration
View Description Hide DescriptionA standard classical model of a socalled rigid twobody collision that employs the dynamic Coulomb friction law to modelfriction is studied. For arbitrary object geometries and initial conditions it is known that the direction of the relative sliding velocity continuously changes during the impact. A (new) exact analytical solution for the relative sliding speed of the two objects in terms of initial conditions and sliding direction is derived. This solution is formulated in terms of a first integral, which is used to rigorously prove that the dynamic Coulomb friction law does not allow either instantaneous sticking or stable sticking to evolve from an initially nonzero except for certain very special cases. The first integral also yields a new procedure for accurately and efficiently computing the final center of mass velocity and the final angular velocity of each of the objects in the model twobody collision. Accurate solutions such as these are essential for analyzing and controlling impacts, which is important, for example, in robot manipulation. Efficient solutions are critically important for producing realtime simulations of rigid twobody collisions.

Towards a classification of Euler–Kirchhoff filaments
View Description Hide DescriptionEuler–Kirchhoff filaments are solutions of the static Kirchhoff equations for elastic rods with circular cross sections. These equations are known to be formally equivalent to the Euler equations for spinning tops. This equivalence is used to provide a classification of the different shapes a filament can assume. Explicit formulas for the different possible configurations and specific results for interesting particular cases are given. In particular, conditions for which the filament has points of selfintersection, selftangency, vanishing curvature or when it is closed or localized in space are provided. The average properties of generic filaments are also studied. They are shown to be equivalent to helical filaments on long length scales.

An exact solution for several charges in classical electrodynamics
View Description Hide DescriptionAn exact solution for an arbitrary number of identical charges that are equally spaced along a circumference and that rotate at constant angular velocity is presented. The solution is valid for any velocity of the charges less than the speed of light and considers the radiation reaction effects as well as the retarded interaction between the charges. The external field that allows this motion consists of a timeindependent electric field tangent to the charges’ orbit and a homogeneous timeindependent magnetic field that points orthogonally to the orbit plane. A detailed analytical study of the total power of radiation associated with this system of charges is carried out, and it is shown that it is in perfect agreement with the energy that the external electric field supplies to the charges. In particular, in the limit when N goes to infinity only static fields remain.
