Volume 25, Issue 1, January 1984
Index of content:

Decomposition of representations into basis representations for the classical groups
View Description Hide DescriptionWe prove decomposition formulae for an arbitrary representation in terms of basis representations for the classical compact Lie groups. Using these decomposition formulae, simple rules are obtained for the product of two arbitrary representations and for the restriction of a representation to a classical subgroup.

On classes of integrable systems and the Painlevé property
View Description Hide DescriptionThe Caudrey–Dodd–Gibbon equation is found to possess the Painlevé property. Investigation of the Bäcklund transformations for this equation obtains the Kuperschmidt equation. A certain transformation between the Kuperschmidt and Caudrey–Dodd–Gibbon equation is obtained. This transformation is employed to define a class of p.d.e.’s that identically possesses the Painlevé property. For equations within this class Bäcklund transformations and rational solutions are investigated. In particular, the sequences of higher order KdV, Caudrey–Dobb–Gibbon, and Kuperschmidt equations are shown to possess the Painlevé property.

Expansions over the ‘‘squared’’ solutions and difference evolution equations
View Description Hide DescriptionThe completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived. It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results. These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scatteringtransform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectraltheory of the operator Λ; (iv) direct calculation of the action‐angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum version of the IST.

A new integral equation for summing Feynman graph series (general scalar Lagrangian case)
View Description Hide DescriptionThe Schwinger parameter formalism is used to derive a new integral equation verified by the ‘‘open’’ four‐point amplitude built from any scalar Lagrangian. This integral equation is a generalization of the one already obtained and studied by the authors in the φ^{3} ladder graph case. One of the main results obtained here is a new representation of the Feynman amplitudes: the so‐called β‐representation, which expresses the Bethe–Salpeter structure of a graph in the Schwinger parameter space. The integrand of the β‐representation satisfies a recurrence relation which is used to sum the perturbation series, and which leads to an integral equation for its sum. The expression of this integral equation is also given in some particular cases (particular values of the invariants, particular classes of graphs, etc.). The Mellin transform of the open amplitude satisfies a similar integral equation which may be used to describe the Regge behavior.

Hamiltonian operators with maximal eigenvalues
View Description Hide DescriptionThe potentials V(x) with a given L ^{ p } norm that maximize the lowest eigenvalue of −Δ+V are characterized.

Stochastic path‐ordered exponentials
View Description Hide DescriptionWe prove convergence of an approximation of the stochastic product integral for conditional Wiener paths to the solution of a certain stochastic integral equation. This is used to establish the Wiener–Itô representation for the kernel of the semigroup exp t Δ_{ A }, where Δ_{ A } =∑_{μ} (∂_{μ} 1+A _{μ})^{2} for functions A _{μ} with values in the space of anti‐Hermitian matrices.

Path integrals in parametrized theories: Newtonian systems
View Description Hide DescriptionConstraints in dynamical systems typically arise either from gauge or from parametrization. We study Newtonian systems moving in curved configuration spaces and parametrize them by adjoining the absolute time and energy as conjugate canonical variables to the dynamical variables of the system. The extended canonical data are restricted by the Hamiltonian constraint. The action integral of the parametrized system is given in various extended spaces: Extended configuration space or phase space and with or without the lapse multiplier. The theory is written in a geometric form which is manifestly covariant under point transformations and reparametrizations. The quantum propagator of the system is represented by path integrals over different extended spaces. All path integrals are defined by a manifestly covariant skeletonization procedure. It is emphasized that path integrals for parametrized systems characteristically differ from those for gauge theories. Implications for the general theory of relativity are discussed.

Quantum energy‐entropy inequalities: A new method for proving the absence of symmetry breaking
View Description Hide DescriptionFor quantum systems we develop a new method, based on a general energy‐entropy inequality, to rule out spontaneous breaking of symmetries. The main advantage of our scheme consists in its clear‐cut physical significance and its new areas of applicability; in particular we can handle discrete symmetry groups as well as continuous ones. Finally a few illustrations are discussed.

Quantum measuring processes of continuous observables
View Description Hide DescriptionThe purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one‐to‐one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.

The Darboux transformation and solvable double‐well potential models for Schrödinger equations
View Description Hide DescriptionThe Darboux transformation, a method used to transform a Schrödinger‐type equation to a Schrödinger equation with a new potential, is discussed. An exactly solvable double‐well potential model for the one‐dimensional Schrödinger equation is obtained.

Transmutation as a minimizing procedure
View Description Hide DescriptionIt is shown formally how transmutation kernels can be characterized via a minimizing procedure. The technique then can be extended to more general operators and transmutations.

Inverse scattering in dimension two
View Description Hide DescriptionThe inverse scattering problem is solved for the two‐dimensional time‐independent Schrödinger equation. That is, the potential is reconstructed from the scattering amplitude, which is assumed to be known for all energies and angles.

Eigenvalues and eigenfunctions associated with the Gel’fand–Levitan equation
View Description Hide DescriptionIt is shown here that the solutions of the Gel’fand–Levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class. The details are sketched for the case of rational reflection coefficients, and carried out for the simplest class of examples.

The causal automorphism of de Sitter and Einstein cylinder spacetimes
View Description Hide DescriptionA well‐known result, due originally to Alexandrov in 1953 and subsequently rediscovered by Zeeman in 1964, states that transformations of Minkowski spacetime which preserve causality are essentially orthochronous Lorentz transformations. In this article, we first exhibit a proof of this result by using a lemma of Zeeman to reduce the proof to another well‐known theorem of Alexandrov involving transformations preserving light speed. Then, by generalizing Zeeman’s lemma and using recent extensions of Alexandrov’s light‐speed theorem, we determine the causal automorphisms of de Sitter and Einstein cylinder spacetimes.

Spherically symmetric solution in the nonsymmetric Kaluza–Klein theory
View Description Hide DescriptionIn this paper we find an exact, static, spherically symmetric solution for the nonsymmetric Kaluza–Klein theory. This solution has the remarkable property of describing ‘‘mass without mass’’ and ‘‘charge without charge.’’ We examine its properties and a physical interpretation.

Solution of multidimensional inverse transport problems
View Description Hide DescriptionFormulas are derived for energy‐dependent, steady‐state, and time‐dependent neutron transport problems, relating the surface neutron fluxes for a convex, homogeneous, three‐dimensional region to the neutron scattering laws that apply within the region. In principle, these formulas can be used to deduce information about the scattering laws.

Symmetric Hadamard series
View Description Hide DescriptionIn a general curved space–time, the requirements that the Feynman Green’s function be symmetric and have the Hadamard form are shown to result in specific constraints on the local behavior of the function. These constraints are solved yielding a general form for the function.

Properties of the Schwinger model
View Description Hide DescriptionWe present all the Wightman functions for an explicit operator solution of the Schwinger model. To understand these better, we study the algebra of fields of this model, representations of this algebra as well as the Hamiltonian. The latter turns out to elucidate the ‘‘confinement’’ of the fermion field. In addition we comment on the renormalization of the theory as well as on the analyticity of the amplitudes in terms of the coupling constant.

A novel mass‐eigenvalue problem for spinors in deSitter space
View Description Hide DescriptionIt is shown that an unambiguous quantum theory of spinors in positively curved deSitter space, based on distinguished coordinates in a Hamiltonian framework, leads to a set of spinors corresponding to unsharp energy but sharp mass defined in a family of novel eigenvalue problems. An example is given in which partly real and partly complex discrete mass spectra come forth.

Vortex properties in first‐ and second‐order formulations of abelian gauge theories
View Description Hide DescriptionProperties of noninteracting vortices in a class of models which generalize the Ginzburg–Landau model of superconductivity are described. Previous results of existence and uniqueness for solutions to the first‐order equations are extended to cover the case in which the gauge photon and the scalar meson become massless, when long range interactions exist. Several properties of the solutions are also discussed. With some assumptions, and with restrictions on the class of models, all finite‐energy solutions of the second‐order equations are shown to be solutions of the first‐order equations. The second‐order equations are formulated in a gauge invariant way, resulting in a second‐order elliptic system of two coupled nonlinear equations, which completely determine all gauge invariant quantities.