The Krichever map,vector bundles over algebraic curves,and Heisenberg algebras

We study the GrassmannianGr _x ⁿ consisting of equivalence classes of rankn algebraic vector bundles over a Riemann surfaceX with an holomorphic trivialization at a fixed pointp. Commutative subalgebras ofgl(n, H ),H being the ring of functions holomorphic on a punctured disc aboutp, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated togl(n, H ₎. One can obtain, by the Krichever map, points ofGr _x ⁿ (and solutions of mcKP) from coveringsf: YX and other geometric data. Conversely for every point ofGr _x ⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr _x ⁿ , an algebraic curve coveringX and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.The research was partially supported by US Army grant DAA L03-87-K-0110 and NSF grant DMS 9106938     

q-Deformation of the Krichever–Novikov Algebra

Using q-operator product expansions between U(1) current fields and the corresponding energy-momentum tensors, we furnish the q-analogues of the generalized Heisenberg algebra and the Krichever–Novikov algebra.     

On nonlocal symmetries for the Krichever–Novikov equation

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever–Novikov equation using the inverse of the fourth-order recursion operator of the latter.     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

A note on the Hohenberg-Kohn theorem

An alternative, concise proof of the general form of Hohenberg-Kohn theorem is presented.     

A note on the interface functions

The functions B₁(τ), D₂(τ), and D₃(τ), which arise in the study of radiative transfer through a planar medium with a relative refractive index different from unity, have been investigated. These functions reduce to exponential integrals when the refractive index of the medium is equal to that of the surroundings. Their behavior has been examined and is presented graphically and in tabular form for various ratios of the interface refractive index. When the interface refractive index ratio is larger than 1.2, these functions can be accurately approximated by an exponential of the form a e^?bτ.     

A note on the Peccei-Quinn symmetry

A modified Peccei-Quinn symmetry proposed by Georgi, which is more suitable for studying the axions, is worked out and the relevant strong CP problem is briefly discussed.     

A note on the emergingn-type superconductors

The effect of doping a Mott-Hubbard insulator to a more-than-half-filled state and the resultant superconduction by electrons are discussed.     

A note on the Wodzicki residue

In this note we explain the relationship of the Wodzicki residue of (certain powers of) an elliptic differential operator P acting on sections of a complex vector bundle E over a closed compact manifold M and the asymptotic expansion of the trace of the corresponding heat operator e^−tP. In the special case of a generalized laplacian Δ and dim M > 2, we thereby obtain a simple proof of the fact already shown in [KW], that the Wodzicki residue res(Δ^−n/2+1) is the integral of the second coefficient of the heat kernel expansion of Δ up to a proportional factor.     

A note on measurement

Grounded on the quantum measurement riddle, a general argument against the universal validity of the superposition principle was recently put forward by Bassi and Ghirardi (Phys. Lett. A 275 (2000) 373). It is pointed out that this argument is valid only within the realm of the philosophy of “objectivistic realism” which is not a necessary part of the foundations of physics, and that recent developments including decoherence theory do account for the appearance of macroscopic objects without resorting to a break of the principle.     

A super soliton connection

Integrable super nonlinear classical partial differential equations are considered. A super sl(2, R) algebra-valued connection 1-form is constructed. It is shown that the curvature 2-form of this super connection vanishes by virtue of the integrable super equations of motion. A super extension of the Ablowitz-Kaub-Newell-Segur scheme is presented and a class of super extensions of the Lax hieararchy and super nonlinear Schrödinger equation are found. The O(N) extension and Bäcklund transformations of the above super equations are also considered.     

A note on the quantum Hamilton-Jacobi formalism

The postulates which have been introduced recently in order to obtain a quantum Hamilton-Jacobi formalism, starting from the classical one, are revised. It is shown that some of them can be replaced by alternative assumptions which happen to be much more natural. The quantum Hamilton-Jacobi equation is obtained as an eigenvalue equation on the quantum wave amplitude.     

A note on the homogeneous balance method

Based on the idea of the homogeneous balance method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear partial differential equations. Some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.     

A note on the wigner distribution function

We rigorously prove that in general it is impossible to obtain a nonnegative phase-space distribution function through averaging a Wigner distribution function (WDF) over the finite phase-space cells. Moreover, we introduce a classification of WDFs according to the possibility of performing such an averaging procedure. Some examples of WDFs are also briefly discussed.     

A note on the exponential kernel approximation

Integral equations, which appear in the study of radiative transfer through a semi- infinite, emitting-absorbing, and isotropically scattering medium, are investigated. Closed form, approximate solutions, which were obtained by using an exponential kernel approximation, are presented and compared with exact solutions.     

A note on the cluster variation method

Kikuchi's cluster variation method (CVM) is reformulated as the truncation of a Möbius inversion. An attempt is made to explicate and simplify the various approaches to the CVM. This formulation makes apparent the connection of the method with other types of cluster approximation. An illustration of the procedure is provided.     

A note on the plasma dispersion function

In the gas laser theory for non-doppler broadening, the plasma dispersion function is often considered to be an antisymmetric function: Z(-μ) = -Z(μ). We shown here that this property is not correct, because the usual integral definition for the plasma dispersion function is not entire in the complex plane.     

A short note on the Aharonov-Bohm effect

We point out that the Aharonov-Bohm effect is a 4-dimensional nonlocal geometric phenomenon. We give two examples which in 3 dimensions appear rather mysterious, but which are easily understood in 4 dimensions. We also discuss why it is integrated effects over fields (potentials) rather than the fields themselves that are important.