Showing papers in "Journal of Mathematical Physics in 1980"

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II

Abstract: We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ordinary differential equations (ODE) of P‐type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P‐type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.

The planar approximation. II

Abstract: The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is illustrated on a simple model coupling two N×N matrices.

A Global Theory of Supermanifolds

Abstract: A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to superfields. This new construction is compared with several pre‐existing definitions of supermanifold and graded manifold; it is shown to include all these definitions and to go beyond them, particularly in admitting the possibility of nontrivial topology in the anticommuting sector. Local differential geometry and potential applications to supergravity are considered.

Dynamics in nonglobally hyperbolic, static space-times

Abstract: Ordinary Cauchy evolution determines a solution of a partial differential equation only within the domain of dependence of the initial data surface. Hence, in a nonglobally hyperbolic space‐time, one does not have fully deterministic dynamics. We show here that for the case of a Klein–Gordon scalar field propagating in an arbitrary static space‐time, a physically sensible, fully deterministic dynamical evolution prescription can be given. If the cosmic censor hypothesis should be overthrown, a prescription of this sort could rescue deterministic physics.

A symmetry approach to exactly solvable evolution equations

Abstract: A method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest‐order derivative. The method, based on the symmetry structure of the equations, is applied to second‐order equations and then to third‐order equations which do not contain a second‐order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third‐order evolution equation which contains the Korteweg‐de Vries (KdV) equation and the modified KdV equation as particular cases.

An improvement of Watson’s theorem on Borel summability

Abstract: Watson’s theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector ‖argz‖<π/2+e, ‖z‖

On the remarkable nonlinear diffusion equation (∂/∂x)[a (u+b)−2(∂u/∂x)]−(∂u/∂t)=0

Abstract: We study the invariance properties (in the sense of Lie–Backlund groups) of the nonlinear diffusion equation (∂/∂x)[C (u)(∂u/∂x)]−(∂u/∂t) =0 We show that an infinite number of one‐parameter Lie–Backlund groups are admitted if and only if the conductivity C (u) =a (u+b)−2 In this special case a one‐to‐one transformation maps such an equation into the linear diffusion equation with constant conductivity, (∂2ū/∂x2)−(∂ū/∂t) =0 We show some interesting properties of this mapping for the solution of boundary value problems

Geometrical reinterpretation of Faddeev–Popov ghost particles and BRS transformations

Abstract: A classical geometrical interpretation of the ghosts fields is presented. BRS rules follow from the Cartan‐Maurer fibration theorem. The statistics of ghosts are explained and the effective quantum Lagrangian is derived without factorizing the volume of the gauge group. Topologically nontrivial ghost configurations are defined.

Space--times with distribution valued curvature tensors

Abstract: A space–time in which in an admissible coordinate system the metric tensor is continuous but has a finite jump in its first and second derivatives across a submanifold will have a curvature tensor containing a Dirac delta function. The support of this distribution may be of three, two, or one dimension or may even consist of a single event. Lichnerowicz’s formalism for dealing with such tensors is modified so as to obtain a formalism in which the Bianchi identities are satisfied in the sense of distributions. The resulting formalism is then applied to the discussion of the Einstein field equations for problems in which the source of the gravitational field is given by a distribution valued stress‐energy tensor. Gravitational shocks are also discussed and their theory is compared with that of high‐frequency gravitational waves given by Y. Choquet‐Bruhat. By considering a class of line sources as obtainable from cylindrical shells by a limiting process, as was proposed by Israel, one may use the distribution formalism developed for hypersurfaces to treat line sources. The line source model proposed by Israel to represent the Kerr metric in the neighborhood of its singular disk is shown to lead to a gravitational mass and angular momentum inconsistent with those of the latter metric. It is proposed to remove this difficulty by changing the assumptions made by Israel concerning the nature of the space–time inside the cylindrical shell which is the support of the distribution in the curvature tensor. The details of the effect of this change are not given in this paper.

Relationships between the group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions

Abstract: We investigate the precise interrelationships between several recently developed solution‐generating techniques capable of generating asymptotically flat gravitational solutions with arbitrary multipole parameters. The transformations we study in detail here are the Lie groups Q and Q of Cosgrove, the Hoenselaers–Kinnersley–Xanthopoulos (HKX) transformations and their SL(2) tensor generalizations, the Neugebauer–Kramer discrete mapping, the Neugebauer Backlund transformations I1 and I2, the Harrison Backlund transformation, and the Belinsky–Zakharov (BZ) one‐ and two‐soliton transformations. Two particular results, among many reported here, are that the BZ soliton transformations are essentially equivalent to Harrison transformations and that the generalized HKX transformation may be deduced as a confluent double soliton transformation. Explicit algebraic expressions are given for the transforms of the Kinnersley–Chitre generating functions under all of the above transformations. In less detail, we also ...

A Lie superalgebraic interpretation of the para‐Bose statistics

Abstract: We show that n pairs of para‐Bose operators generate the classical simple orthosymplectic Lie superalgebra osp(1,2n) ≡B (0,n). The creation and annihilation operators are negative and positive root vectors, respectively, and they span a basis in the odd part of B (o,n).

Group actions on principal bundles and invariance conditions for gauge fields

Abstract: Invariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification of group actions on bundles as automorphisms projecting to an action on a base manifold with a sufficiently regular orbit structure is given in terms of group homorphisms and a generalization of Wang’s theorem classifying invariant connections is derived. Illustrative examples on compactified Minkowski space are given.

Particle statistics from induced representations of a local current group

Abstract: Representations of the nonrelativistic current group S‐K are studied in the Gel’fand–Vilenkin formalism, where S is Schwartz’ space of rapidly decreasing functions, and K is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of KF (the stability group of a point F∈S′) which factors through the permutation group SN. Starting from a quasi‐invariant measure μ concentrated on a K orbit Δ in S′, together with a suitable representation of KF for F∈Δ, sufficient conditions are developed for inducing a representation of S‐K. The Hilbert space for the induced representation consists of square‐integrable functions on a covering space of Δ, which transform in accordance with a representation of KF. The Bose and Fermi N‐particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representat...

Schrödinger spectral problems with energy–dependent potentials as sources of nonlinear Hamiltonian evolution equations

Abstract: We develop a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrodinger spectral problems whose potential functions depend on the spectral parameter.

Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion

Abstract: We report several important additions to our original discussion of Ermakov systems. First, we show how to derive the Ermakov system from more general equations of motion. Second, we show that there is a general nonlinear superposition law for Ermakov systems. Also, we give explicit examples of the nonlinear superposition law. Finally, we point out that any ordinary differential equation can be included in many Ermakov systems.

A homogeneous Hilbert problem for the Kinnersley–Chitre transformations

Abstract: A homogeneous Hilbert (Riemann) problem (HHP) is introduced for carrying out the Kinnersley–Chitre transformations of the set V of all axially symmetric stationary vacuum spacetimes, and the spacetimes which are like the axially symmetric stationary ones except that both Killing vectors are spacelike. A proof, which is independent of the Kinnersley–Chitre formalism, establishes that the HHP transforms the potential (for certain closed self‐dual 2 forms) F0(x, t) of any given member of V into the potential F (x, t) of another member of V. Two illustrative examples involving the Minkowski space F0(x, t) are given. The representation used for the Geroch group K, the singularities and gauge of the potentials, and possible applications of the HHP are discussed.

Factorization of operators I. Miura transformations

Abstract: The method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is here extended to some third‐order scattering operators, and transformations between several fifth‐order nonlinear evolution equations are derived. Further applications are discussed.

Inverse scattering. I. One dimension

Abstract: This paper presents two new methods of reconstructing an underlying potential in the one‐dimensional Schrodinger equation from a given S matrix. One of these methods is based on a Gel’fand–Levitan equation, the other on a Marchenko equation. A sequel of this paper will treat the three‐dimensional case by similar methods.

U(N) Integrals, 1/N, and the De Wit–’t Hooft anomalies

Abstract: Formulas for the evaluation of all U(N) integrals are derived. Tables display the results for integrands involving up to six U’s and six U°’s. The complete pole structure of De Wit–’t Hooft anomalies is unveiled. The effects of 1/N2 corrections and De Wit–’t Hooft anomalies on two‐dimensional U(N) lattice gauge theories in the strong coupling 1/N expansion is discussed.

Equilibrium states for mean field models

Abstract: We rigorously characterize the KMS and the limiting Gibbs states for mean field models. As an application we prove the convergence of the Gibbs states for the Dicke Maser model in the infinite volume limit.

Representations and properties of para‐Bose oscillator operators. I. Energy position and momentum eigenstates

Abstract: Para‐Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation Dα of the para‐Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation Dα, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.

Path integral in the representation of SU(2) coherent state and classical dynamics in a generalized phase space

Abstract: Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two‐dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.

Positive quantum joint distributions

Abstract: We demonstrate the existence of positive phase space density functions which yield the quantum mechanical marginal distributions of position and momentum.

N‐dimensional instantons and monopoles

Abstract: The possibility of finding solutions of the instanton and monopole types to gauge field theories on arbitrary even and odd dimensional Euclidean manifolds respectively is investigated. Suitable boundary conditions for both types are given, and new self‐duality criteria are developed, for gauge field theories on N‐dimensional manifolds (N?5) which are also endowed with new Action and Lagrangian densities.

Classical supersymmetric particles

Abstract: The dynamics of classical spinning particles is studied from the point of view of gauge supersymmetry. The central idea is that the natural way of introducing intrinsic spin degrees of freedom into a physical system is to take the square root of the Hamiltonian generators of the system without spin, which is equivalent to rendering the system gauge supersymmetric. This is accomplished by describing the spin degrees of freedom by means of ’’anticommuting c‐numbers’’ (odd Grassman algebra elements) and relying on Dirac’s theory of constrained Hamiltonian systems. The requirement of gauge supersymmetry fixes completely the action principle and leaves neither room nor need for a d h o c subsidiary conditions on the relative direction of the spin and the velocity as in the more traditional treatments. Both massive and massless particles free and in interaction with electromagnetic and gravitational fields are discussed. It is found that there exists a supergauge in which the spin tensor of a massive particle in a gravitational field is transported in parallel but the particle does not follow a geodesic. Massless particles on the other hand have the property of possessing a supergauge where their helicity is conserved and in which at the same time the worldline is a geodesic. Special attention is paid to the meaning and properties of the supergauge transformations. The main aspects of that discussion are applicable to more complicated systems such as supergravity. In particular phenomena such as necessity of invoking the equations of motion to close the gauge are analyzed.

Inverse scattering. II. Three dimensions

Abstract: Assuming that a scattering amplitude, given as a function of the energy and the directions of the incident and scattered particles, is associated with a local potential without spherical symmetry via the Schrodinger equation in three space dimensions, this potential is uniquely reconstructed by two methods. One is based on a generalization of the Marchenko equation; the other, on a generalization of the Gel’fand–Levitan equation.

The use of anticommuting variable integrals in statistical mechanics. II. The computation of correlation functions

Abstract: Integrals over anticommuting variables are use to rewrite partition functions as fermionic field theories. The method is used to solve the two‐dimensional Ising model, the planar close‐packed dimer problems, and the free‐fermion eight vertex model.

Resonances, scattering theory, and rigged Hilbert spaces

Abstract: The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.

A matrix representation of the translation operator with respect to a basis set of exponentially declining functions

Abstract: The matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert space L2(R3) are given in closed form as functions of the displacement vector. The basis functions are composed of an exponential, a Laguerre polynomial, and a regular solid spherical harmonic. With this formalism, a function which is defined with respect to a certain origin, can be ’’shifted’’, i.e., expressed in terms of given functions which are defined with respect to another origin. In this paper we also demonstrate the feasibility of this method by applying it to problems that are of special interest in the theory of the electronic structure of molecules and solids. We present new one‐center expansions for some exponential‐type functions (ETF’s), and a closed‐form expression for a multicenter integral over ETF’s is given and numerically tested.

Classification of real simple Lie superalgebras of classical type

Abstract: Finite‐dimensional simple Lie superalgebras (also called Z2‐graded Lie algebras) over an algebraically closed field of characteristic zero were classified in 1976. All simple Lie superalgebras over the reals, whose Lie subalgebra is reductive, are determined here up to isomorphism. As is the theory of simple Lie algebras, this is done by classifying the involutive semimorphisms of the complex Lie superalgebras. One sees in particular that the real form of the Lie subalgebra completely determines the real form of the Lie superalgebra.