Showing papers in "Journal of Mathematical Physics in 1978"

A Simple model of the integrable Hamiltonian equation

Abstract: A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrodinger equation and the so‐called Harry Dym equation turn out to be particular examples.

An exact solution for a derivative nonlinear Schrödinger equation

Abstract: A method of solution for the ’’derivative nonlinear Schrodinger equation’’ i q t =−q x x ±i (q*q 2) x is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

A unified treatment of null and spatial infinity in general relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity

Abstract: A new definition of asymptotic flatness in both null and spacelike directions is introduced. Notions relevant to the null regime are borrowed directly from Penrose’s definition of weak asymptotic simplicity. In the spatial regime, however, a new approach is adopted. The key feature of this approach is that it uses only those notions which refer to space–time as a whole, thereby avoiding the use of a initial value formulation, and, consequently, of a splitting of space–time into space and time. It is shown that the resulting description of asymptotic flatness not only encompasses the essential physical ideas behind the more familiar approaches based on the initial value formulation, but also succeeds in avoiding the global problems that usually arise. A certain 4‐manifold—called Spi (spatial infinity) —is constructed using well‐behaved, asymptotically geodesic, spacelike curves in the physical space–time. The structure of Spi is discussed in detail; in many ways, Spi turns out to be the spatial analog of I. The group of asymptotic symmetries at spatial infinity is examined. In its structure, this group turns out to be very similar to the BMS group. It is further shown that for the class of asymptotically flat space–times satisfying an additional condition on the (asymptotic behavior of the ’’magnetic’’ part of the) Weyl tensor, a Poincare (sub‐) group can be selected from the group of asymptotic symmetries in a canonical way. (This additional condition is rather weak: In essence, it requires only that the angular momentum contribution to the asymptotic curvature be of a higher order than the energy–momentum contribution.) Thus, for this (apparently large) class of space–times, the symmetry group at spatial infinity is just the Poincare group. Scalar, electromagnetic and gravitational fields are then considered, and their limiting behavior at spatial infinity is examined. In each case, the asymptotic field satisfies a simple, linear differential equation. Finally, conserved quantities are constructed using these asymptotic fields. Total charge and 4‐momentum are defined for arbitrary asymptotically flat space–times. These definitions agree with those in the literature, but have a further advantage of being both intrinsic and free of ambiguities which usually arise from global problems. A definition of angular momentum is then proposed for the class of space–times satisfying the additional condition on the (asymptotic behavior of the) Weyl tensor. This definition is intimately intertwined with the fact that, for this class of space–times, the group of asymptotic symmetries at spatial infinity is just the Poincare group; in particular, the definition is free of super‐translation ambiguities. It is shown that this angular momentum has the correct transformation properties. In the next paper, the formalism developed here will be seen to provide a platform for discussing in detail the relationship between the structure of the gravitational field at null infinity and that at spatial infinity.

Entropy production for quantum dynamical semigroups

Abstract: In analogy to the phenomenological theory of irreversible thermodynamics we define the entropy production for an arbitrary quantum dynamical semigroup with a stationary state. We prove that the entropy production is convex and positive and that the entropy production is a measure of dissipativity of the semigroup. The entropy production is used to prove the approach to equilibrium and to classify the stationary states of semigroups arising in the weak coupling limit.

Presymplectic manifolds and the Dirac-Bergmann theory of constraints

Abstract: We present an algorithm which enables us to state necessary and sufficient conditions for the solvability of generalized Hamilton‐type equations of the form ι (X) ω=α on a presymplectic manifold (M,ω) where α is a closed 1‐form. The algorithm is phrased in the context of global infinite‐dimensional symplectic geometry, and generalizes as well as improves upon the local Dirac–Bergmann theory of constraints. The relation between our algorithm and the geometric constraint theory of Śniatycki, Tulczyjew, and Lichnerowicz is discussed.

Solitons and rational solutions of nonlinear evolution equations

Abstract: Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods. In this note specific attention is directed at the Korteweg–de Vries equation. However, the methods used are quite general and apply to most nonlinear evolution equations with the isospectral property, including certain multidimensional equations. In the latter case, nonsingular, algebraically decaying, soliton solutions can be constructed.

Entropy of an n‐system from its correlation with a k‐reservoir

Abstract: Let a random pure state vector be chosen in nk‐dimensional Hilbert space, and consider an n‐dimensional subsystem’s density matrix P. P will usually be close to the totally unpolarized mixed state if k is large. Specifically, the rms deviation of a probability from the mean value 1/n is [(1−1/n2)/(kn +1)]1/2. ’’Random’’ refers to unitarily invariant Haar measure.

Asymptotic behavior of group integrals in the limit of infinite rank

Abstract: We show that in the limit N→∞ integrals with respect to Haar measure of products of the elements of a matrix in SO(N) approach corresponding moments of a set of independent Gaussian random variables Similar asymptotic forms are obtained for SU(N) and Sp(N) An application of these results to Wilson’s formulation of lattice gauge theory is briefly considered

Transmission problems for the Helmholtz equation

Abstract: A straightforward treatment of these problems is given which appears to avoid many of the previously encountered difficulties. Admittedly some generality is lost by assuming that the various associated parameters ki are not space dependent within their respective domains of definition, Di. Nevertheless, by means of the approach offered here, such problems can be analyzed in just one function space; more general existence and uniqueness theorems can be obtained; there is no need to regularize the operators involved; and, above all, the solutions can be expressed in terms of certain boundary integral equations which, computationally, offer good prospects.

Generalization of Dirac’s monopole to SU2 gauge fields

Abstract: Dirac’s monopole is generalized to SU2 gauge fields in five‐dimensional flat space or four‐dimensional spherical space. The generalized fields have SO5 symmetry. The method used is related to the concept of orthogonal gauge fields which is developed in an appendix. The angular momenta operators for the SO5 symmetrical fields are given.

Exact solitary ion acoustic waves in a magnetoplasma

Abstract: It is shown that finite amplitude ion acoustic solitary waves propagating obliquely to an external magnetic field can occur in a plasma.

Nonuniqueness in the inverse scattering problem

Abstract: The inverse scattering problem consists of determining the functional form of a scattering potential given the scattering matrix A (k0s, k0s0) for all scattering directions s and one or more values of the wave vector k0s0 In this paper it is shown that within the framework of the first Born approximation the inverse scattering problem as defined above does not possess a unique solution It is also shown that within the framework of exact (potential) scattering theory the problem does not admit a unique solution given only the scattering matrix for a single fixed value of the wave vector k0s0 as data The final section in the paper considers scattering experiments using incident fields other than plane waves and where knowledge of the scattered field at all points exterior to the scattering volume is available as data It is found that, within the framework of exact scattering theory, the data generated by any single such experiment is not sufficient to uniquely specify the scattering potential while, wit

Symmetries of the stationary Einstein–Maxwell equations. IV. Transformations which preserve asymptotic flatness

Abstract: We give a series of transformations βk, k=0,1,... which may be used to generate new stationary axially‐symmetric vacuum solutions from ones already known. These transformations have the important property of preserving asymptotic flatness. As one example of their use, we show how to generate the Kerr metric from Schwarzschild. As a second example, we generate a new five‐parameter vacuum solution which contains the δ=2 Tomimatsu–Sato solution as a special case.

On invariant integration over SU(N)

Abstract: We give a graphical algorithm for evaluation of invariant integrals of polynomials in SU(N) group elements. Such integrals occur in strongly coupled lattice gauge theory. The results are expressed in terms of totally antisymmetric tensors and Kronecker delta symbols.

Conformal extensions of the Galilei group and their relation to the Schrödinger group

Abstract: Various authors have considered a conformal extension CG0 of the Galilei group which in some sense is the nonrelativistic limit of the conformal extension of the Poincare group, and have also established an invariance group for the free‐particle Schrodinger equation, the ’’Schrodinger group.’’ Here we establish the most general conformal extension CG of the Galilei group, which is found to be identical to the group of the most general coordinate transformations that permit the use of noninertial frames of reference and of curvilinear coordinates in Galilei‐invariant theories, which was considered by one of us some time ago, and is a gauge group containing a number of arbitrary functions. Both CG0 and the Schrodinger group are subgroups of CG containing the Galilei group, but otherwise they do not overlap. The Hamilton–Jacobi and Schrodinger equations for particles which are free or interact via inverse‐square potentials are shown to be invariant under the Schrodinger group, and a further invariance of the...

A new form of the Mayer expansion in classical statistical mechanics

Abstract: New expressions are given for the expansion coefficients in the Mayer expansion (and thus the virial expansion). These promise to be useful in applications, as well as provide a simple rigorous proof of the convergence of the Mayer series and some of its properties.

The Use of Anticommuting Integrals in Statistical Mechanics. 1.

Abstract: Integrals over anti commuting variables are used to rewrite partition functions as fermionic field theories. In particular, the method is applied to the two-dimensional Ising and dimer models. LBL-8217 Supported by the High Energy Physics Division of the United States Department of Energy. -2

Inequalities and uncertainty principles

Abstract: Sobolev inequalities give lower bounds for quantum mechanical Hamiltonians. These inequalities are derived from commutator inequalities related to the Heisenberg uncertainty principle.

On the positivity of energy in general relativity

Abstract: We show that the positive energy argument of Geroch for time‐symmetric initial data sets can be generalized to general initial data sets

Self‐avoiding random walks: Some exactly soluble cases

Abstract: We use the exact renormalization group equations to determine the asymptotic behavior of long self‐avoiding random walks on some pseudolattices. The lattices considered are the truncated 3‐simplex, the truncated 4‐simplex, and the modified rectangular lattices. The total number of random walks C_n, the number of polygons P_n of perimeter n, and the mean square end to end distance 〈R^2_n〉 are assumed to be asymptotically proportional to μ^nn^(γ−1), μ^nn^(α−3), and n^(2ν) respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν.

On the interaction of the type λx2/(1+gx2)

Abstract: The ground state and the first two excited state energy levels for the interaction of the type λx2/(1+gx2) have been calculated nonperturbatively as the eigenvalues of the one‐dimensional Schrodinger operator defined by Au=−u′′+x2u+λx2u/(1+gx2). The Ritz variational method in combination with the Givens–Householder algorithm has been used for numerical computations.

Sequences of Z2⊗Z2 graded Lie algebras and superalgebras

Abstract: Applying methods similar to those used for classical Lie superalgebras (Z2 graded algebras), we construct sequences of Z2⊕Z2 graded Lie superalgebras. In this way one obtains the spl(m,n,r,s), osp(m,n,r,s), P1(m,r), P3(m,n), ospP3(m,n), P1,2(m), and Q (m) series. We also give series of Z2⊕Z2 graded Lie algebras. Closed forms for superdeterminants and determinants of Z2⊕Z2 graded matrices are presented.

Dense electron‐gas response at any degeneracy

Abstract: An exact expression for the linear response function of the dense electron gas valid at any temperature is worked out in the ring (RPA) approximation. The T=0 and T=∞ limits reproduce the already known results. It is used to explain the longitudinal oscillations and the screening around a test charge. The latter is either Thomas–Fermi‐like or Friedel‐like according to the values of the parameters.

Coherent electromagnetic waves in pair‐correlated random distributions of aligned scatterers

Abstract: Recent results for the corresponding scalar problem are generalized to coherent electromagnetic waves in random distribution of pair‐correlated obstacles (aligned or averaged over alignment). Proceeding essentially as before, we obtain dispersion equations by averaging the vector–dyadic functional equation relating the multiple and single scattered amplitudes of the obstacles. In general, for aligned nonradially symmetric scatterers, the resulting bulk indices of refraction specify anisotropic media; the anisotropy arises either from the scatterers’ properties (physical parameters or shape, or both) or from their distribution, or from both. The illustrations include both isotropic and anisotropic cases, and the explicit results generalize earlier ones.

The three‐dimensional convolution of reduced Bessel functions and other functions of physical interest

Abstract: A method for evaluating convolution integrals over rather general functions is suggested, based on the analytical evaluation of convolution integrals over functions BMν,L(r) = (2/π)1/2rL+νKν (r) YML(ϑ,φ), which are products of modified Bessel functions of the second kind Kν(r), regular solid spherical harmonics rLYML(ϑ,φ), and powers rν.

Tensor fields invariant under subgroups of the conformal group of space‐time

Abstract: This work is concerned with the characterization of tensor fields in (compactified) Minkowski space which are invariant under the action of subgroups of the conformal group. The general method for determining all invariant fields under the smooth action of a Lie group G on a manifold M is given, both in global and in local form. The maximal subgroups of the conformal group are divided into conjugacy classes under the Poincare group and the most general fields of 1‐forms, 2‐forms, symmetric (0,2) tensors and scalar densities which are invariant under representatives of each class (as well as certain other subgroups) are then determined. The results are then discussed from the viewpoint of physical interpretation (as, e.g., electromagnetic fields, metric tensors, etc.) and applicability; in particular, for studies of spontaneously or otherwise broken conformal invariance.

Path integrals for solving some electromagnetic edge diffraction problems

Abstract: Electromagnetic edge diffraction problems involving parallel half‐planes are traditionally attacked by the Wiener–Hopf technique, or asymptotically for a large wavenumber (k→∞) by ray‐optic techniques. This paper reports a novel method in which the electromagnetic wave equation is first converted to a heat equation via the Laplace transform. The heat equation together with the original boundary condition is next solved approximately in terms of a path integral over the Wiener measure. For several examples involving two parallel half‐planes, the path integral is evaluated explicitly to yield an asymptotic solution of order k0 for the field on the incident shadow boundary. Those solutions agree with the ones derived by traditional techniques, but are obtained here in a much simpler manner. In other examples involving multiple half‐planes, the use of a path integral leads to new solutions. We have not succeeded, however, in generating higher‐order terms beyond k0 in the asymptotic solution by path integrals.

An exact solution to Einstein’s equations with a stiff equation of state

Abstract: A solution to the equations of general relativity is given which is spherically‐symmetric and nonstatic with an inhomogeneous density profile ρ and a pressure p given by the stiff equation of state p=ρc2. The solution may be of use in representing collapsed astrophysical systems or the early stages of an inhomogeneous cosmology.

Conformal two‐structure as the gravitational degrees of freedom in general relativity

Abstract: In this paper, we suggest that what we shall call the conformal 2‐structure may, in an appropriate coordinate system, serve to embody the two gravitational degrees of freedom of the Einstein (vacuum) field equations. The conformal 2‐structure essentially gives information concerning the manner in which a family of 2‐surfaces is embedded in a 3‐surface. We show that, formally at least, this prescription works for the exact plane and cylindrical gravitational wave solutions, for the double‐null and null‐timelike characteristic initial value problems, and for the usual Cauchy spacelike initial value problem. We conclude with a preliminary consideration of a two‐plus‐two breakup of the field equations aimed at unifying these and other initial value problems; and a discussion of some aspirations and remaining problems of this approach.

Elementary construction of graded Lie groups

Abstract: We show how the definitions of the classical Lie groups have to be modified in the case where Grassmann variables are present. In particular we construct the general linear, the special linear and the orthosymplectic graded Lie groups. Special attention is paid to the question of how to formulate an adequate ’’unitarity condition.’’