Showing papers in "Journal of Mathematical Physics in 1979"

A method for determining a stochastic transition

Abstract: A number of problems in physics can be reduced to the study of a measure‐preserving mapping of a plane onto itself. One example is a Hamiltonian system with two degrees of freedom, i.e., two coupled nonlinear oscillators. These are among the simplest deterministic systems that can have chaotic solutions. According to a theorem of Kolmogorov, Arnol’d, and Moser, these systems may also have more ordered orbits lying on curves that divide the plane. The existence of each of these orbit types depends sensitively on both the parameters of the problem and on the initial conditions. The problem addressed in this paper is that of finding when given KAM orbits exist. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of nearby periodic orbits. The relation between KAM surfaces and periodic orbits has been explored extensively here by the numerical computation of a particular mapping. An important part of this procedure is the introduction of two quantities, the residue and the mean residue, that permit the stability of many orbits to be estimated from the extrapolation of results obtained for a few orbits. The results are distilled into a series of assertions. These are consistent with all that is previously known, strongly supported by numerical results, and lead to a method for deciding the existence of any given KAM surface computationally.

Two‐dimensional lumps in nonlinear dispersive systems

Abstract: Two‐dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two‐dimensional nonlinear Schrodinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N‐soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.

Generalized Lie algebras

Abstract: The generalized Lie algebras, which have recently been introduced under the name of color (super) algebras, are investigated. The generalized Poincare–Birkhoff–Witt and Ado theorems hold true. We discuss the so‐called commutation factors which enter into the defining identities of these algebras. Moreover, we establish a close relationship between the generalized Lie algebras and ordinary Lie (super) algebras.

A guiding center Hamiltonian: A new approach

Abstract: A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B (x,y) ? is studied. Lie transforms are used to carry out the perturbation expansion.

Note on the stability of the Schwarzschild metric

Abstract: It is shown that the standard arguments for the stability of the Schwarzschild metric can be made into a rigorous proof that the numerical values of linear perturbations of Schwarzschild must remain uniformly bounded for all time.

Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory

Abstract: We study numerically a method of summation of divergent series based on an order dependent mapping. We consider the example of a simple integral, of the ground‐state energy of the anharmonic oscillator and of the critical exponents of φ34 field theory. In the case of the simple integral convergence can be rigorously proven, while in the other examples we can only give heuristic arguments to explain the properties of the transformed series. For the anharmonic oscillator we have compared our results to an accurate numerical solution (10−23) of the Schrodinger equation. For the critical exponents we have verified the consistency of our results with those obtained before from methods using a Borel transformation.

The classical limit of quantum nonspin systems

Abstract: The classical limit of operators X belonging to any compact Lie algebra g is computed. If X∈g, the classical limit in the representation ΓΛ, whose highest weight is Λ, is lim ΓΛ(X/N) =Σsig (fi,X,Ω), where the limit is taken as N→∞, the sum runs from i=1 to r=rank g, Λ=Σμifi,fi are the highest weights of the r fundamental representations of g,si=lim μi/N, and g (fi,X,Ω) is the expectation value of X with respect to the coherent states ‖fi, Ω〉 in the representation Γfi. Examples and applications are given.

Symmetries of the stationary Einstein–Maxwell equations. VI. Transformations which generate asymptotically flat spacetimes with arbitrary multipole moments

Abstract: A new series of transformations is presented for generating stationary axially symmetric asymptotically flat vacuum solutions of Einstein’s equations. The application requires only algebraic manipulations to be performed. Several examples are given of new stationary axisymmetric solutions obtained in this way. It is conjectured that the transformations, applied to the genral Weyl metric, can be used to generate systematically all stationary metrics with axial symmetry.

Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras

Abstract: We present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two‐index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest‐weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.

Scattering theory and polynomials orthogonal on the unit circle

Abstract: The techniques of scattering theory are used to investigate polynomials orthogonal on the unit circle. The discrete analog of the Jost function, which has been shown to play an important role in the theory of polynomials orthogonal on a segment of the real line, is defined for this system and its properties are investigated. The relation between the Jost function and the weight function is discussed. The techniques of inverse scattering theory are developed and used to obtain new asymptotic formulas satisfied by the polynomials. A set of sum rules satisfied by the coefficients in the recurrence relaxation is exhibited. Finally, Szego’s theorem on Toeplitz determinants is proved using the recurrence formulas and the Jost function. The techniques of inverse scattering theory are used to find the correction terms.

A critique of the major approaches to damping in quantum theory

Abstract: We examine the two major approaches that have been suggested for the quantum mechanical treatment of the damped motion of a particle as a one‐body problem. These are the linear, but time dependent, Kanai Hamiltonian, and the more recent nonlinear potentials which have been introduced to simulate the damping force. The most important criticism that has been leveled at the Kanai Hamiltonian is that its solutions seem to violate the uncertainty relations. We show that this Hamiltonian actually represents a particle of variable mass, whose classical behavior is identical to that of a damped particle of constant mass. But quantum mechanically, its changing mass does lead to unphysical behavior when misinterpreted as a constant mass particle. So this Hamiltonian cannot directly describe a constant mass damped quantum particle. The nonlinear model has been interpreted in terms of the hydrodynamical analogy of quantum theory, and a well behaved decaying wavepacket solution has been produced. However we generalize this result to produce solutions that ’’decay’’ to arbitrarily high energy. Thus it is not clear that this model specifically treats dissipation. Rather it seems to seek out any stationary state. At any rate, its physical interpretation is obscure at present. However we show, by analyzing the physical problem of damping at low energies, that one can modify the Kanai Hamiltonian to eliminate its unphysical features, so that this modified Kanai Hamiltonian can in fact be interpreted as representing a constant mass damped particle with physically reasonable solutions.

Shear‐free gravitational collapse

Abstract: Spherically symmetric perfect fluids are studied under the restriction of shear‐free motion. All solutions of the field equations are found by solving a single second order nonlinear equation containing an arbitrary function. It is shown that this arbitrary function is a geometric invariant, E, which measures the gravitational field energy, and it is shown that E=const generates all the homogeneous density solutions. An improved proof is given for the nonexistence of any one‐parameter equation of state. A number of exact solutions are presented and discussed.

One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature

Abstract: We compute exactly the one particle reduced density matrix ρ (r) of a system of impenetrable bosons in one dimension at zero temperature. We do this by relating ρ (r) to a certain double scaling limit of the transverse correlation function of the one‐dimensional spin 1/2 X–Y model. We study the asymptotic behavior of ρ (r) for large r. This expansion contains oscillatory terms which arise due to the intrinsic quantum mechanical nature of the problem. We use these results to discuss the analytic structure of the momentum density function n (k).

Deformations of gauge groups. Gravitation

Abstract: This is a review, and an attempt at completion, of the ’’Gupta program,’’ the ultimate goal of which is either to show that Einstein’s theory of gravitation is the only self‐consistent field theory of interacting, massless, spin‐2 particles in flat space or to discover interesting alternatives. It is useful to notice that the gauge group of general relativity is a deformation (in a mathematically precise sense) of the gauge group associated with the massless, spin‐2 free field. The uniqueness of Einstein’s theory depends on the stability of its gauge group with respect to a class of differentiable deformations. A generalized Gupta program for massless fields of arbitrary spins is proposed.

Quantum theory of infinite component fields

Abstract: The quantum theory of the infinite component SO(4,2) fields is formulated as a model for relativistic composite objects. We discuss three classes (timelike, lightlike, and spacelike) of physical solutions to a general class of infinite component wave equations. These solutions provide a definite physical interpretation to infinite component wave equations and are obtained by reducing SO(4,2) with respect to its orthogonal, pseudo‐orthogonal, and Euclidean subgroups. The analytic continuations among these solutions are established. In the nonrelativistic limit the timelike physical states exactly reduce to the Schrodinger solution for the hydrogen atom—the simplest composite object. The wave equations for the three classes are studied in two different realizations. In one case the equations describe a three‐dimensional internal Kepler motion with a discrete and a continuous energy spectrum and in the other case the equations describe a four‐dimensional internal oscillatory motion with attractive as well as repulsive potentials. It is found that the Kustaanheimo–Steifel transformation of classical mechanics exactly relates these two internal motions also in the quantum case. Thus a completely relativistic theory of composite systems is established for which the internal dynamics is the generalization of the nonrelativistic two‐body dynamics.

Group theory of the interacting boson model of the nucleus

Abstract: Recently Arima and Iachello proposed an interacting boson model of the nucleus involving six bosons, five in a d and one in an s state The most general interaction in this model can then be expressed in terms of Casimir operators of the following chains of subgroups of the fundamental group U(6): U(6) ⊆U(5) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆O(6) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆SU(3) ⊆O(3) ⊆O(2) To determine the matrix elements of this interaction in, for example, a basis characterized by the irreducible representations of the first chain of groups, then we only need to evaluate the matrix elements of the Casimir operators of O(6) and SU(3) in this basis as the others are already diagonal in it Using results of a previous publication for the basis associated with U(5) ⊆O(5) ⊆O(3), we obtain the matrix elements of the Casimir operators of O(6) and SU(3) Furthermore, we obtain explicitly the transformation brackets between states characterized by irreducible representations of the first two chains of groups Numerical programs are being developed for these matrix elements from the relevant reduced 3j symbols for the O(5) ⊆O(3) chain of groups that were programmed previously

On conserved quantities in general relativity

Abstract: Recently, definitions of total 4‐momentum and angular momentum of isolated gravitating systems have been introduced in terms of the asymptotic behavior of the Weyl curvature (of the underlying space–time) at spatial infinity. Given a space–time equipped with isometries, on the other hand, one can also construct conserved quantities using the presence of the Killing fields. Thus, for example, for stationary space–times, the Komar integral can be used to define the total mass, and, the asymptotic value of the twist of the Killing field, to introduce the dipole angular momentum moment. Similarly, for axisymmetric space–times, one can obtain the (’’z‐component’’ of the) total angular momentum in terms of the Komar integral. It is shown that, in spite of their apparently distinct origin, in the presence of isometries, quantities defined at spatial infinity reduce to the ones constructed from Killing fields. This agreement reflects one of the many subtle aspects of Einstein’s (vacuum) equation.

Normal form for mirror machine Hamiltonians

Abstract: A systematic algorithm is developed for performing canonical transformations on Hamiltonians which govern particle motion in magnetic mirror machines. These transformations are performed in such a way that the new Hamiltonian has a particularly simple normal form. From this form it is possible to compute analytic expressions for gyro and bounce frequencies. In addition, it is possible to obtain arbitrarily high order terms in the adiabatic magnetic moment expansion. The algorithm makes use of Lie series, is an extension of Birkhoff’s normal form method, and has been explicitly implemented by a digital computer programmed to perform the required algebraic manipulations. Application is made to particle motion in a magnetic dipole field and to a simple mirror system. Bounce frequencies and locations of periodic orbits are obtained and compared with numerical computations. Both mirror systems are shown to be insoluble, i.e., trajectories are not confined to analytic hypersurfaces, there is no analytic third i...

Noether’s theorem, time‐dependent invariants and nonlinear equations of motion

Abstract: Noether’s theorem is applied to a Lagrangian for a system with nonlinear equations of motion. Noether’s theorem leads to a time‐dependent constant of the motion along with an auxiliary equation of motion. Special cases of this invariant have been used to quantize the time‐dependent harmonic oscillator. We also discuss the solution of the original equations of motion in terms of the solutions to the auxiliary equation.

The classification of all H spaces admitting a Killing vector

Abstract: We show that all H spaces (self‐dual solutions of the complex Einstein vacuum equations) that admit (at least) one Killing vector may be gauged in such a way as to be divided into only five types, characterized by the type of equation which determines their potential function. In four of these types we show that this knowledge is sufficient to reduce the requirement of being an H space to a linear equation whose solutions are well known. The fifth case is reduced considerably and a large class of special solutions is given.

Reduction of the two‐dimensional O(n) nonlinear σ‐model

Abstract: We reduce the field equations of the two‐dimensional O(n) nonlinear σ‐model to relativistic O(n−2) covariant differential equations involving n−2 scalar fields.

On the complete integrability of the stationary, axially symmetric Einstein equations

Abstract: A linear eigenvalue problem in the spirit of Lax is constructed for the nonlinear differential equations describing stationary, axially symmetric Einstein spaces. In suitable variables these equations yield a generalization of the well‐known sine‐Gordon equation. The similarity of the system to the nonlinear σ model is pointed out.

Exact solution of a time‐dependent quantal harmonic oscillator with damping and a perturbative force

Abstract: The problem of a quantal harmonic oscillator with damping and a time‐dependent frequency acted on by a time‐dependent perturbative force is exactly solved. The wavefunctions are found in Schrodinger representation using the theory of explicitly time‐dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.

Construction of Lie algebras and Lie superalgebras from ternary algebras

Abstract: Ternary algebras are algebras which close under suitable triple products. They have been shown to be building blocks of ordinary Lie algebras. They may acquire a deep physical meaning in fundamental theories given the important role played by Lie (super) algebras in theoretical physics. In this paper we introduce the concept of superternary algebras involving Bosers and Fermi variables. Using them as building blocks, we give a unified construction of Lie algebras and superalgebras in terms of (super) ternary algebras. We prove theorems that must be satisfied for the validity of this construction, which is a generalization of Kantor’s results. A large number of examples and explicit constructions of the Lie algebras An, Bn, Fn, Fn, F4, F6, F7, F8, and Lie superalgebras A (m,n), B (m,n), D (m,n), P (n), Q (n) are given. We speculate on possible physical applications of (super) ternary algebras.

A new approach to inhomogeneous cosmologies: Intrinsic symmetries. I

Abstract: After briefly discussing systematic ways of studying inhomogeneous cosmologies, we propose the technique of ’’intrinsic symmetries,’’ in which restrictions are placed on submanifolds of space–time. This leads to a broad classification of inhomogeneous cosmologies, and to an associated specialization diagram. By enforcing additional restrictions we obtain a further useful specialization.

The inverse problem for random sources

Abstract: The problem of deducing the statistical structure of a localized random source ρ (r) of the reduced wave equation from measurements of the field external to the source is addressed for the case when the measurements yield the autocorrelation function of the field at all pairs of points exterior to the source volume and the quantity to be determined is the source’s autocorrelation function Rρ(r1,r2) =〈ρ* (r1) ρ (r2) 〉.This problem is shown to be equivalent to that of determining Rρ from the autocorrelation function of the field’s radiation pattern and is found, in general, not to admit a unique solution due to the possible existence of nonradiating sources within the source volume. Notable exceptions are the class of delta correlated (incoherent) sources whose intensity profiles are shown to be uniquely determined from the data and the class of quasihomogeneous sources whose coherence properties can be determined if their intensity profiles are known and vice versa.

Rigid body motions, space curves, prolongation structures, fiber bundles, and solitons

Abstract: The dynamics of a nonlinear string of constant length represented by a helical space curve may be studied through a consideration of the motion of an arbitrary rigid body along it. The resulting set of compatibility equations is shown to result in the class of nonlinear evolution equations solvable through the two component inverse scattering phenomenology. A class of pseudopotentials and prolongation structures follow naturally due to the intrinsic group structure of the phenomenon. This leads to an identification of the underlying fiber bundle structure and connection forms. Thus a unified picture emerges for a class of soliton possessing evolution equations.

The non‐Markovian relaxation process as a ‘‘contraction’’ of a multidimensional one of Markovian type

Abstract: A new approach for obtaining the Fokker–Planck equation to be associated with the generalized Langevin equation is discussed. By using the Mori expansion of the ’’memory kernel,’’ it is shown that any information of interest may be provided by a suitable multidimensional Fokker–Planck equation of Markovian type. A suitable ’’contraction’’ process, furthermore, enables us to find the same two‐point conditional probability as the one recently obtained by Fox. This approach may be useful to overcome the Markov approximation which is present in the stochastic Liouville equation theory.

Coupled inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer

Abstract: To the second order in metric and the first order in equations of motion in the local coordinates of an accelerated rotating observer, the inertial effects and gravitational effects are simply additive. To look into the coupled inertial and gravitational effects, we derive the third‐order expansion of the metric and the second‐order expansion of the equations of motion in local coordinates. Besides purely gravitational (purely curvature) effects, the equations of motion contain, in this order, the following coupled inertial and gravitational effects: redshift corrections to electric, magnetic, and double‐magnetic type curvature forces; velocity‐induced special relativistic corrections; and electric, magnetic, and double‐magnetic type coupled inertial and gravitational forces. An example is provided with a static observer in the Schwarzchild spacetime.

Linear response theory revisited. II. The master equation approach

Abstract: We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iL0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.