Volume 39, Issue 1, January 1998
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


A Dirac sea and thermodynamic equilibrium for the quantized threewave interaction
View Description Hide DescriptionThe classical version of the threewave interaction models the creation and destruction of waves; the quantized version models the creation and destruction of particles. The quantum threewave interaction is described and the Bethe ansatz for the eigenfunctions is given in closed form. The Bethe equations are derived in a rigorous fashion and are shown to have a thermodynamic limit. The Dirac sea of negative energy states is obtained as the infinite density limit. Finite particle/hole excitations are determined and the asymptotic relation of energy and momentum is obtained. The Yang–Yang functional for the relative free energy of finite density excitations is constructed and is shown to be convex and bounded below. The equations of thermal equilibrium are obtained.

Proof of Serban’s conjecture
View Description Hide DescriptionWe prove Serban’s conjecture which greatly simplifies the expression of the advanced singleparticle Green function in the Calogero–Sutherland model. The importance of proving this conjecture is that it reorganizes the form factor in terms of two dimensional Coulomb gas correlators and confirms the possible existence of a bosonization procedure for this system.

Aharonov–Bohm effect with δtype interaction
View Description Hide DescriptionA quantum particle interacting with a thin solenoid and a magnetic flux is described by a fiveparameter family of Hamilton operators, obtained via the method of selfadjoint 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.

On eigenvalues in gaps for perturbed magnetic Schrödinger operators
View Description Hide DescriptionWe study Schrödinger operators with a gap in the essential spectrum, perturbed by either a decreasing electric potential or a decreasing magnetic field; in both cases the strength of the perturbation is measured by a coupling constant Here we are mainly interested in the asymptotic behavior (as ) of certain counting functions for the eigenvalues that are produced by the perturbation inside the spectral gap. The case where we perturb by a potential can be handled using current technology, even if contains a fixed magnetic background. For perturbations by magnetic fields, however, we require rather strong assumptions—like exponential decay of the perturbations—to obtain a lower bound on the counting function. To gain some additional intuition, we use separation of variables in the closely related model of a Schrödinger operator with constant magnetic field in perturbed by a rotationally symmetric magnetic field that decays at infinity.

On infinitedimensional algebras of symmetries of the selfdual Yang–Mills equations
View Description Hide DescriptionInfinitedimensional algebras of hidden symmetries of the selfdual Yang–Mills equations are considered. A currenttype algebra of symmetries and an affine extension of conformal symmetries introduced recently are discussed using the twistor picture. It is shown that the extended conformal symmetries of the selfdual Yang–Mills equations have a simple description in terms of Ward’s twistor construction.

Quantum sources and a quantum coding theorem
View Description Hide DescriptionWe define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon–McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented.

Moving frames hierarchy and BF theory
View Description Hide DescriptionWe show that the onedimensional projection of Chern–Simons gauged nonlinear Schrödinger model is equivalent to an Abelian gauge field theory of a continuum Heisenberg spin chain. In such a theory, the matter field has geometrical meaning of coordinates in tangent plane to the spin phase space, while the gauge symmetry relates to rotation in the plane. This allows us to construct the infinite hierarchy of integrable gauge theories and related magnetic models. To each of them a invariant gauge fixing constraint of nonAbelian BF theory is derived. The corresponding moving frames hierarchy is obtained and the spectral parameter is interpreted as a constantvalued statistical gauge potential constrained by the 1cocycle condition.

Twobody Dirac equations for general covariant interactions and their coupled Schrödingerlike forms
View Description Hide DescriptionWe present new and useful “external potential” forms of the twobody Dirac equations of constraint dynamics for combined scalar, vector, pseudoscalar, pseudovector, and tensor interactions. These equations have potential applications in twobody problems for bound states in meson spectroscopy and phase shift analysis in nucleon–nucleon scattering. Toward this end, we derive their coupled Schrödingerlike forms using matrix techniques and obtain the corresponding radial equations to these forms from scalar and vector spherical harmonic decompositions.

Unitary transformation approach to the exact solution for a class of timedependent nonlinear Hamiltonian systems
View Description Hide DescriptionBy performing unitary transformations on the timedependent Schrödinger equation, the exact solution for a class of nonlinear Hamiltonian systems is obtained and it is shown that these timedependent problems are related to the associated time independent problems. In addition, the evolution operator is derived. Nonadiabatic Berry’s phase is calculated on the basis of the exact solution. The theory is applied to some illustrative examples.

Abelian Chern–Simons theory. I. A topological quantum field theory
View Description Hide DescriptionWe give a construction of the Abelian Chern–Simons gauge theory from the point of view of a dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas that have been previously explored in connection to the nonAbelian Chern–Simons theory [J. Diff. Geom. 33, 787–902 (1991); Topology32, 509–529 (1993)]. We formulate the topological quantum field theory in terms of the category of extended 2 and 3manifolds introduced in a preprint by Walker in 1991 and prove that it satisfies the axioms of unitary topological quantum field theories formulated by Atiyah [Publ. Math. Inst. Hautes Etudes Sci. Pans 68, 175–186 (1989)].

Abelian Chern–Simons theory. II. A functional integral approach
View Description Hide DescriptionFollowing Witten, [Commun. Math. Phys. 21, 351–399 (1989)] we approach the Abelian quantum Chern–Simons (CS) gauge theory from a Feynman functional integral point of view. We show that for 3manifolds with and without a boundary the formal functional integral definitions lead to mathematically proper expressions that agree with the results from the rigorous construction [J. Math. Phys. 39, 170–206 (1998)] of the Abelian CS topological quantum field theory via geometric quantization.

Infinite and finite Gleason’s theorems and the logic of indeterminacy
View Description Hide DescriptionIn the first half of the paper I prove Gleason’s lemma: Every nonnegative frame function on the set of rays in 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 (nonnegative 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).

On the energy spectrum of the hydrogen atom in a photon field. I
View Description Hide DescriptionIn 1930 Weisskopf and Wigner gave an account, based on the Maxwell–Schrödingerequations, of the natural spectral line broadening of radiation emitted by a hydrogen atom. Their calculations were based on an approximation involving certain singlephoton transitions in the perturbation series for the solutions of these equations. In Part I of this series of papers the exact expressions for both the line shift and the line broadening are obtained from the Maxwell–Diracequations in such a way that the Weisskopf–Wigner results appear as a second order approximation. The Maxwell–Dirac Hamiltonian for the coupled fields is first shown to admit a complex analytic dilation in the energy variables. The Fredholm–Born series for the resolvent is shown to converge uniformly when certain highenergy cutoff factors are included in the interaction and the photons are given a small mass. The series is then rearranged to show that the spectrum of the modified dilated Hamiltonian, which consists of a complete set of complex eigenvalues, thresholds, and branch cuts, is only a slight perturbation of the known spectrum of the dilated Hamiltonian for the uncoupled fields. The real part of the shift of each complex eigenvalue then accounts for the spectral line shift, and the complex part accounts for the associated line broadening. Finally, the implications for the scattering matrix and the various phenomena of resonance scattering are discussed. In Part II of this series these results are shown to remain valid when the cutoff factors and the photon mass are removed.

Periodic orbit theory analysis of a continuous family of quasicircular billiards
View Description Hide DescriptionWe compute the Fourier transform of the quantum mechanical energy level density for the problem of a particle in a twodimensional circular infinite well (or circular billiard) as well as for several special generalizations of that geometry, namely the halfwell, quarterwell, and the circular well with a thin, infinite wall along the positive axis (hereafter called a circular well plus baffle). The resulting peaks in plots of versus are compared to the lengths of the classical closed trajectories in these geometries as a simple example of the application of periodic orbit (PO) theory to a billiard or infinite well system. We then solve the Schrödinger equation for the general case of a circular well with infinite walls both along the positive axis and at an arbitrary angle Θ (a circular “slice”) for which the halfwell (Θ=π), quarterwell (Θ=π/2), and circular well plus baffle are then all special cases. We perform a PO theoryanalysis of this general system and calculate for many intermediate values of Θ to examine how the peaks in attributed to periodic orbits change as the quasicircular wells are continuously transformed into each other. We explicitly examine the transitions from the halfcircular well to the circle plus baffle case (halfwell to quartercircle case) as Θ changes continuously from π to 2π (from π to π/2) in detail. We then discuss the general Θ→0 limit, paying special attention to the cases where , as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic orbit theoryanalysis is of great benefit in understanding and visualizing the increasingly complex pattern of closed orbits as Θ→0.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


New conformal gauging and the electromagnetic theory of Weyl
View Description Hide DescriptionA new eightdimensional conformal gauging solves the auxiliary field problem and eliminates unphysical size change from Weyl’s electromagnetic theory. We derive the Maurer–Cartan structure equations and find the zero curvature solutions for the conformal connection. By showing that every oneparticle Hamiltonian generates the structure equations we establish a correspondence between phase space and the eightdimensional base space, and between the action and the integral of the Weyl vector. Applying the correspondence to generic flat solutions yields the Lorentz force law, the form and gauge dependence of the electromagnetic vector potential and minimal coupling. The dynamics found for these flat solutions applies locally in generic spaces. We then provide necessary and sufficient curvature constraints for general curved eightdimensional geometries to be in 1–1 correspondence with fourdimensional Einstein–Maxwell space–times, based on a vector space isomorphism between the extra four dimensions and the Riemannian tangent space. Despite part of the Weyl vector serving as the electromagnetic vector potential, the entire class of geometries has vanishing dilation, thereby providing a consistent unified geometrictheory of gravitation and electromagnetism. In concluding, we give a concise discussion of observability of the extra dimensions.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


On the second law of thermodynamics and contact geometry
View Description Hide DescriptionIn this paper we consider the second law of thermodynamics for a dissipative system and its symmetry property in terms of contact geometry. We first show that the inaccessibility condition of Caratheodory and the assumption of semipositive definite property of the dissipative energy are equivalent to Clausius’ inequality. The inaccessibility condition then gives rise to a generalized Gibbs relation (GGR). By means of the GGR a 1form ω can be defined such that the zero of ω reproduces the GGR. Such 1form ω has the property and The integral surface of the GGR is an dimensional 1graph space (Legendre submanifold) of a 1jet space where is the base space of with thermodynamic variables as its coordinates. The dimensional equipped with the 1form ω is also called a contact bundle where the intensive thermodynamic variables are considered as the contact elements to at every Next we construct an isovector field such that the inaccessibility condition is invariant under the contact transformations generated by Finally, suppose under some specific assumptions the dynamical equations of the thermodynamic variables can be approximated by the flow equations of a vector field on We can lift to such that the 1graph space as well as the inaccessibility condition are preserved under the contact transformations generated by

Informationtheoretical derivation of an extended thermodynamical description of radiative systems
View Description Hide DescriptionA radiative equation of the Cattaneo–Vernotte type is derived from information theory and the radiative transfer equation. The equation thus derived is a radiative analog of the equation that is used for the description of hyperbolic heat conduction. It is shown, without recourse to any phenomenological assumption, that radiative transfer may be included in a natural way in the framework of extended irreversible thermodynamics (EIT).

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Conditions of Cayley’s type for ellipsoidal billiard
View Description Hide DescriptionThe billiard system within an ellipsoid in threedimensional space is considered. We describe periodical trajectories of that mechanical system deriving analytical condition of Cayley’s type for generalized Poncelet’s theorem, using methods of algebrogeometric integration.

Heisenberg chain systems from compact manifolds into
View Description Hide DescriptionIn this paper, it is shown that the initial value problems of Heisenberg Spin Chain Systems with external magnetic field from compact Riemannian manifolds into admit global weak solutions.

 RELATIVITY AND GRAVITATION


Spacetime defects: Domain walls and torsion
View Description Hide DescriptionThe theory of distributions in nonRiemannian spaces is used to obtain exact static thin domain wallsolutions of EinsteinCartan equations of gravity. Curvature singularities are found while Cartan torsion is given by Heaviside functions. Weitzenböck planar walls are characterized by torsion singularities and zero curvature. It is shown that Weitzenböck static thin domain walls do not exist exactly as in general relativity. The global structure of Weitzenböck nonstatic torsion walls is investigated.
