Index of content:
Volume 36, Issue 1, January 1995

Diffeomorphism cohomology in Beltrami parametrization. II. The 1‐forms
View Description Hide DescriptionThe 1‐form diffeomorphism cohomologies within a local conformal Lagrangianfield theory model built on a two dimensional Riemann surface with no boundary is studied. The case of scalar matter fields is considered and the complex structure is parametrized by Beltrami differential. The analysis is first performed at the classical level, and then we improve the quantum extension, introducing the current in the Lagrangian dynamics, coupled to external source fields. It is shown that the anomalies which spoil the current conservations take origin from the holomorphic region of the external fields, and only the differential spin 1 and 2 currents (as well as their c.c) could be anomalous.

A phase‐space technique for the perturbation expansion of Schrödinger propagators
View Description Hide DescriptionA perturbation theory for Schrödinger and heat equations that is based on phase‐space variables is developed. The Dyson series representing the evolution kernel is described in terms of two basic classical quantities: the free classical motion along flat space geodesics and the Green function for the Jacobi operator in phase space. Further, for problems with Abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel. Connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals. This type of application is discussed for the case of Schrödinger and heat equations with external electromagnetic fields. Detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time Schwinger–DeWitt expansion. Finally it is shown how to apply this phase‐space method so that it incorporates a recently proposed covariant perturbation theory.

Mixed states with positive Wigner functions
View Description Hide DescriptionThe Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low‐dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.

Bound state variational principle employing a discontinuous trial function
View Description Hide DescriptionA new variational principle for the bound state energy eigenvalues of a quantum system is presented. In it, not only ∇ψ(r) but also ψ(r) itself can be discontinuous across a surface. That is, the trial ψ(r) can have both a kink and a jump across a surface. This allows the use of different basis functions in different regions of space without having to impose any matching conditions across the surface. All parameters appearing in the principle can be variational parameters; none are needed to help invoke any continuity conditions on the trial function. Although the principle is not a definite (minimum) one, good results are obtained in several examples.

Exact calculation of the ground‐state single‐particle Green’s function for the 1/r ^{2} quantum many‐body system at integer coupling
View Description Hide DescriptionThe ground‐state single‐particle Green’s function describing hole propagation is calculated exactly for the 1/r ^{2} quantum many‐body system at integer coupling. The result is in agreement with a recent conjecture of Haldane.

Generalizations of the Wess–Zumino–Witten model and Ferretti–Rajeev model to any dimensions: CP‐invariance and geometry
View Description Hide DescriptionThe Wess–Zumino–Witten model in two dimensions is generalized to one in any even dimensions and the Ferretti–Rajeev model in three dimensions to one in any odd dimensions. In this connection some properties of fermion number (topological) currents in odd dimensions are also discussed.

Charge transfer model and three body‐scattering
View Description Hide DescriptionThe asymptotics of total scattering cross sections for a three‐body quantum system with two heavy particles are studied when the masses tend to infinity. It is assumed that the initial channel is a two‐cluster channel such that a light particle and a heavy one form a bound state and the relative velocity of another heavy particle to the pair is fixed. It is shown that the limits exist and they can be expressed in terms of transition probabilities which appear in the scattering theory for the charge transfer models.

Density correlation inequalities analogous to Bell’s
View Description Hide DescriptionThe usual Bell‐type inequalities involve correlations between observables. Inequalities of the Bell‐type are presented which involve correlations between the probabilities associated with states and which can be used as a test for hidden variables in quantum mechanics independent of the Bell inequalities. The inequalities are universal in the sense that they are valid for all separable metric state spaces, as opposed to the usual Bell inequalities whose specific form is a function of the state space of the candidate classical model. The inequalities are harder to prove than the usual Bell‐type inequalities because no essential restrictions on the state space are made.

Passage from quantum systems with continuous spectrum to quantum Poisson processes on Hilbert modules
View Description Hide DescriptionThe present article is devoted to the explanation of the irreversible behavior of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system, whose Hamiltonian has a continuous spectrum, interacting with a reservoir and studies the limits of quantities related to the whole compound system. A macroscopic equation is obtained for the limit of the compound system, which is a quantum stochastic differential equation of Poisson type on some Hilbert module (no longer a space) and whose coefficients are uniquely determined by the one‐particle Hamiltonian of the original system and whose driving noises are the creation, annihilation, and number (or gauge) processes living on the Fock module over this module.

Can elementary quantum mechanics explain the Aharonov–Bohm effect?
View Description Hide DescriptionIt is shown that a satisfactory treatment of the Aharonov–Bohm problem requires no departure from the basic principles of ordinary quantum mechanics, a fact not entirely clear in the literature. The discussion ends up with the same formalism as originally proposed by Aharonov and Bohm, in agreement with experimental facts, although in a way that forces no alteration in the traditional interpretation of vector potentials.

Energy levels of double‐well potentials in a three‐dimensional system
View Description Hide DescriptionThe renormalized inner product perturbation technique is used to compute the energy levels for a double‐well potential V(x,y,z;Z ^{2},λ)=−Z ^{2}[x ^{2}+y ^{2}+z ^{2}] +λ[a _{ xx } x ^{4}+a _{ yy } y ^{4}+a _{ zz } z ^{4} +2a _{ xy } x ^{2} y ^{2}+2a _{ xz } x ^{2} z ^{2} +2a _{ yz } y ^{2} z ^{2}] in the three‐dimensional system. Results are produced for a wide range of parameters Z ^{2}, λ and of state numbers (n _{ x },n _{ y },n _{ z }). Comparison is made with results obtained by other means.

Lanczos potential for some type D space–times
View Description Hide DescriptionA Lanczos potential of any non‐null orbit type D solution of the Einstein–Maxwell equations with nonsingular aligned electromagnetic field is obtained. In the case of the Kerr–Newman solution the Lanczos potential is also explicitly given.

Statistical mechanics of hard spheres
View Description Hide DescriptionA theoreticalequation of state for a system of hard spheres is proposed. It is based on the ideas of the scaled‐particle theory that has been developed by Reiss et al. [J. Chem. Phys. 31, 369 (1959)]. In this paper, their method is generalized by expanding the function G(y,r) in a set of functions t _{ i }(r) so that the low‐density behavior is described exactly, not only for asymptotically large values of r, but for the full range of r values. As a consequence the virial coefficients B _{3} and B _{4} come out exactly in the approach to the scaled particle theory. For the compressibility factor, a Padé‐like expression is obtained that gives exact values for B _{3} and B _{4} and reproduces B _{5}, B _{6}, B _{7}, and B _{8} to within 1.3%, 1.4%, 0.74%, and 5.5%, respectively. Furthermore, the equation of state agrees with the simulation data up to high densities, where the fluid is metastable. An expression for the surface tension is also found.

Integral expressions of the derivatives of the small‐angle scattering correlation function
View Description Hide DescriptionThe integral expressions of the derivatives of the small‐angle scatteringcorrelation function are obtained in terms of the parametric equations of the interphase surfaces. For each derivative, its integral expression is the surface average of a loop integral of a vectorial field that is the only quantity depending on the derivative order. The field relevant to the nth derivative is obtained by letting a suitable differential operator act (n‐2) times on the field relevant to the second derivative. Both the differential operator and the last vectorial field are given in terms of the interface parametric equations. Starting from these expressions, the limiting values of the derivatives of the correlation function as r → 0 are studied by a procedure similar to the one used by Kirste and Porod [Kolloid‐Z. 184, 1 (1962)] and Wu and Schmidt [J. Appl. Crystallogr. 4, 224 (1971)]. In this way, the conjecture put forth by Wu and Schmidt that, when the interfaces are sufficiently smooth, all even‐order derivatives of the correlation function are null at the origin is proven to be true.

Topological aspects of spin and statistics in nonlinear sigma models
View Description Hide DescriptionSpin and statistics for topological solitons in nonlinear sigma models are studied using topological methods based on the classical configuration space. Taking as space the connected d‐manifold X, and considering nonlinear sigma models with the connected manifoldM as target space, topological solitons are given by elements of π_{ d }(M). Any topological soliton α∈π_{ d }(M) determines a quotient Stat_{ n }(X,α) of the group of framed braids with n strands on X, such that a framed braid determines a contractible path in the n‐soliton sector of the configuration space if and only if its image in Stat_{ n }(X,α) is the identity. In particular, when M=S ^{2}, as in the O(3) nonlinear sigma model with Hopf term, and α∈π_{2}(S ^{2}) is a generator, we compute that Stat_{ n }(R ^{2},α)=Z, while Stat_{ n }(S ^{2},α)=Z _{2n }. Thus one expects the phase exp(iθ) for interchanging two solitons of type α on S ^{2} to satisfy the constraint θ=kπ/n, k∈Z, when n such solitons are present.

The Hamiltonian structures of the super‐KP hierarchy associated with an even parity super‐Lax operator
View Description Hide DescriptionThe even parity super‐Lax operator is considered for the supersymmetric KP hierarchy of the form L=D ^{2}+∑^{∞} _{ i=0} u _{ i−2} D ^{−i+1} and two Hamiltonian structures are obtained following the standard method of Gel’fand and Dikii. It is observed that the first Hamiltonian structure is local and linear whereas the second Hamiltonian structure is nonlocal and nonlinear among the superfields appearing in the Lax operator. Their connections with the super‐w _{∞}algebra are discussed briefly.

Tests of integrability of the supersymmetric nonlinear Schrödinger equation
View Description Hide DescriptionIn this paper various conventional tests of integrability are applied to the supersymmetric nonlinear Schrödinger equation. It is found that a matrix Lax pair exists and that the system has the Painlevé property only for a particular choice of the free parameters of the theory. It is also shown that the second Hamiltonian structure generalizes to superspace only for these values of the parameters. It has not been possible to construct a zero curvature formulation of the equations based on OSp(2‖1). However, this attempt yields a nonsupersymmetric fermionic generalization of the nonlinear Schrödinger equation which appears to possess the Painlevé property.

Asymptotic eigenvalue degeneracy for a class of three‐dimensional Fokker–Planck operators
View Description Hide DescriptionIn this paper, the spectrum of a class of three‐dimensional Fokker–Planck operators is studied. Asymptotic eigenvalue degeneracy is proved when the diffusion constant ε goes to zero. A lower bound on the gap between the ground state and the first eigenvalue in each l channel is given.

The Korteweg–de Vries hierarchy and long water‐waves
View Description Hide DescriptionBy using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface‐waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion, we show that the Korteweg–de Vries hierarchy equations do play a role in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.

Tau functions, Korteweg–de Vries flows, and supersymmetric quantum mechanics
View Description Hide DescriptionIn this article the connection between the τ functions of the Korteweg–de Vries (KdV) hierarchy and reflectionless potentials with an arbitrary number of bound states constructed using supersymmetric quantum mechanics is studied. It is shown how the underlying KdV flows can be extracted from the construction based on supersymmetric quantum mechanics.