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Showing papers in "Journal of Mathematical Physics in 1982"


Journal ArticleDOI
TL;DR: In this paper, the authors provide necessary and sufficient conditions for several observables to have a joint distribution and show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables.
Abstract: We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no‐hidden‐variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.

328 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show that all real, Euclidean self-dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; these are the generators of a translational or a rotational symmetry.
Abstract: Using the formalism of complex H‐spaces, we show that all real, Euclidean self‐dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; in Kahler coordinates these are the generators of a translational or a rotational symmetry. We give explicit forms both for the Killing vectors and for the constraint on the Kahler potential function Ω which allows for such a Killing vector. In the translational case we show how all such spaces are determined by the general solution of the three‐dimensional, flat Laplace’s equation and how these are related to the multi‐Taub–NUT metrics of Gibbons and Hawking. In the rotational case we simplify the equation determining Ω, but this is not sufficient to obtain the general solution.

278 citations


Journal ArticleDOI
TL;DR: In this paper, a Gauss-Bonnet type identity in Riemann-Cartan geometry was derived, where (−g) 1/2eμνλρ (R μν ε + (1/2) C α μν C αλν) = ∂μ (−(−g)1/ 2eμπλρ C μνλ ϵ ε C αμν C ε), where ε is the torsion tensor.
Abstract: We derive a new Gauss–Bonnet type identity in Riemann‐Cartan geometry: (−g)1/2eμνλρ (R μνλρ + (1/2) C α μν C αλν) = ∂μ (−(−g)1/2eμνλρ C μνλρ), where C α μν is the torsion tensor.

267 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the Henon-Heiles Hamiltonian in the complex time plane and showed that the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable.
Abstract: The solutions of the Henon–Heiles Hamiltonian are investigated in the complex time plane. The use of the ’’Painleve property,’’ i.e., the property that the only movable singularities exhibited by the solution are poles, enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable. Special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes. These boundaries have a remarkable self‐similar structure whose form changes as a function of the nonlinear coupling.

233 citations


Journal ArticleDOI
TL;DR: In this article, the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt and perturbation analysis is applied to this entire form, in all of its components.
Abstract: The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt. The perturbation analysis is applied to this entire form, in all of its components.

203 citations


Journal ArticleDOI
TL;DR: In this article, an algorithmic method was developed for investigating the transformation properties of second-order equations of Painleve type, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painlesve equations II-VI, with different parameters.
Abstract: An algorithmic method is developed for investigating the transformation properties of second‐order equations of Painleve type. This method, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painleve equations II–VI, with different parameters. These transformations easily generate rational and other elementary solutions of the equations. The relationship between Painleve equations and certain new equations quadratic in the second derivative of Painleve type is also discussed.

199 citations


Journal ArticleDOI
TL;DR: For a classical Hamiltonian H=(1/2)p2+V(q,t) with an arbitrary time-dependent potential V(qs,t), exact invariants that can be expressed as series in positive powers of ǫ p, I(qp,p,t)=∑∞n=0pnfn(qs),t, are examined in this article.
Abstract: For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.

181 citations


Journal ArticleDOI
TL;DR: In this paper, a complete classification of the unitary irreducible representations of the Poincare group is provided, and generalized Foldy-Wouthuysen transformations are constructed to connect the physical UIRs with covariant field theories in three dimensions.
Abstract: We provide a complete classification of the unitary irreducible representations of the (2+1)‐dimensional Poincare group. We show, in particular, that only two types of ’’spin’’ are available for massless field theories. We also construct generalized Foldy–Wouthuysen transformations which connect the physical UIR’s with covariant field theories in three dimensions.

168 citations


Journal ArticleDOI
TL;DR: In this paper, completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by either using the Bargmann representation or by using the kq======representation which was introduced by J. Zak.
Abstract: It is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the Bargmann representation or by using the kq representation which was introduced by J. Zak. In this paper both methods are considered, in particular, in connection with expansions of generalized functions in what are called Gabor series. The setting consists of two spaces of generalized functions (tempered distributions and elements of the class S*) which appear in a natural way in the context of the Bargmann transform. Also, a thorough mathematical investigation of the Zak transform is given. This paper contains many comments and complements on existing literature; in particular, connections with the theory of interpolation of entire functions over the Gaussian integers are given.

158 citations


Journal ArticleDOI
TL;DR: In this article, the spectral theory for quasiperiodic sine and sinh-Gordon equations is given, and the relation between the ingredients in the inverse spectral solution of the periodic sine-Gordon equation and physical characteristics of sine•Gordon waves is emphasized.
Abstract: A summary of the spectral theory for quasiperiodic sine‐ and sinh‐Gordon equations is given. Analogies with whole‐line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine‐Gordon equation and physical characteristics of sine‐Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.

129 citations


Journal ArticleDOI
TL;DR: In this article, a general expression for the susceptibility and conductivity of weakly interacting particles is derived for the many-body linear response expressions, where the diagonal parts depend on the scattering processes, and the closed expressions for general two-body collisions are cumbersome.
Abstract: The many‐body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron‐phonon interaction. For elastic collisions simple closed forms result. For general two‐body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams‐Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.

Journal ArticleDOI
TL;DR: In this paper, an algorithm for obtaining an hereditary symmetry (the generalized squared eigenfunction operator) from a given isospectral eigenvalue problem is presented. But this method is applied to the n×n eigen value problem considered by Ablowitz and Haberman.
Abstract: We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared‐eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.

Journal ArticleDOI
TL;DR: In this paper, the exact solutions of Einstein's equation that represent static, axisymmetric black holes distorted by an external matter distribution are obtained and the structure of the structure is examined.
Abstract: All exact solutions of Einstein’s equation that represent static, axisymmetric black holes distorted by an external matter distribution are obtained. Their structure—local and global—is examined. The Hawking temperature is derived and laws of thermodynamics given for both the total system of black hole and external matter and the black hole considered as a single system. The evolution, induced by Hawking radiation, of distorted black holes is discussed.

Journal ArticleDOI
TL;DR: In this article, the relationship between the light reflected from a surface and its shape was investigated, and it was shown that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point.
Abstract: In this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape. Particular emphasis is given to eikonal equations. Two results are presented. First, we prove that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point. Using this result, we show that a restricted form of this equation has exactly two solutions. These results have application in scanning electron microscopy.

Journal ArticleDOI
TL;DR: In this article, a method is proposed to obtain the dynamics of a system which only makes use of the group law. But the method is applied to the free-particle dynamics and the harmonic oscillator.
Abstract: A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free‐particle dynamics and the harmonic oscillator.

Journal ArticleDOI
TL;DR: In this article, a new procedure for deriving integrable Hamiltonians and their constants of the motion is introduced, called the truncation program, which is a generalization of the Whittaker program.
Abstract: A new procedure for deriving integrable Hamiltonians and their constants of the motion is introduced. We term this procedure the truncation program. Integrable Hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e. The relationship between this program and the Whittaker program in two degrees of freedom is discussed. Integrable Hamiltonians occurring in the Whittaker program (a generalization of Whittaker’s work) possess constants of the motion which are polynomials in the momentum coordinates. Many previously known integrable Hamiltonians are derived. A new family of integrable double resonance Hamiltonians and a new family of integrable Hamiltonians of the form (p21+p22)/2+V(q1, q2) are derived.

Journal ArticleDOI
TL;DR: In this article, a new Stokes' line can arise when previously defined Stokes’ lines cross, and a new formulation of the WKB problem is given to justify the new Stoke' lines.
Abstract: The WKB theory for differential equations of arbitrary order or integral equations in one dimension is investigated. The rules previously stated for the construction of Stokes’ lines for Nth‐order differential equations, N⩾3, or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path. This paradox is resolved by the demonstration that new Stokes’ lines can arise when previously defined Stokes’ lines cross. A new formulation of the WKB problem is given to justify the new Stokes’ lines. With the new Stokes’ lines, the asymptotic forms can be shown to be independent of path. In addition, the WKB eigenvalue problem is formulated, and the global dispersion relation is shown to be a functional of loop integrals of the action.

Journal ArticleDOI
TL;DR: In this paper, the authors give formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations; among them, the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and the Kupershmidt equation.
Abstract: Using a bi‐Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth‐ and seventh‐order nonlinear partial differential equations; among them, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ‖t‖→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.

Journal ArticleDOI
TL;DR: In this article, the authors exploit a fruitful one-to-one correspondence between the Lie group branching SU (N+M) and the supergroup branchings SU(N/M) for decomposition of representations of supergroups into representations of subgroups.
Abstract: The decomposition of representations of supergroups into representations of subgroups is needed in practical applications. In this paper we set up and exploit a fruitful one‐to‐one correspondence between the Lie group branching SU (N+M)⊇SU(N)⊗SU(M)⊗U(1) and the supergroup branchings SU(N/M)⊇SU(N)⊗SU(M)⊗U(1) and SU(N 1+N 2/M 1+M 2)⊇SU(N 1/M 1) ⊗SU(N 2/M 2)⊗U(1). A simple and useful prescription is discovered for obtaining the SU(N/M) branching rules from those of SU(N+M) for any representation. A large class of examples, sufficient for many physical applications we can foresee, are explicitly worked out and tabulated.

Journal ArticleDOI
TL;DR: In this paper, one-dimensional electromagnetic and elastic inverse problems for media with discontinuous material properties are formulated, and the measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem.
Abstract: One‐dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties. In addition, an impedence mismatch between source and medium is allowed. The measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem. The solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media.

Journal ArticleDOI
TL;DR: In this article, it was shown that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten's equation and that becomes constant at infinity.
Abstract: We prove that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten’s equation and that becomes constant at infinity. Thus we fill a gap in Witten’s arguments on the nonnegativity of the total mass of an isolated system, when measured at spatial infinity. We also include a review of Witten’s argument.

Journal ArticleDOI
TL;DR: In this article, the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed, which describes long internal gravity waves in a stratified fluid of finite depth and reduces to the Korteweg-de Vries equation as shallow water limit and the Benjamin-Ono equation as deep water limit.
Abstract: The inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed. The equation describes long internal gravity waves in a stratified fluid of finite depth, and reduces to the Korteweg–de Vries equation as shallow water limit and the Benjamin–Ono equation as deep water limit. Both limits of the method and novel aspects of the theory are also discussed.

Journal ArticleDOI
TL;DR: In this article, two integrals which appear in the study of the relativistic Bose gas are analyzed and the complete lowtemperature and hightemperature expansions are computed, as well as their complete low-temperature expansion.
Abstract: Two integrals which appear in the study of the relativistic Bose gas are analyzed. The complete low‐temperature and high‐temperature expansions are computed.

Journal ArticleDOI
TL;DR: In this article, the authors quantize non-canonically a system of two nonrelativistic point particles, interacting via a harmonic potential, in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations.
Abstract: Following the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase‐space variables are quantized in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations. As a result, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. In particular, the distance between the constituents is preserved in time and can take at most four different values. The orbital momentum is either zero or one (in units ℏ/2). The operators of the coordinates do not commute with each other and, therefore, the position of any one of the constituents cannot be localized; the particles are smeared with a certain probability in a finite space volume, which moves together with the center of mass. In the limit ℏ→0 the constituents ‘‘fall’’ into their center of mass and the composite system behaves as a free point particle.

Journal ArticleDOI
TL;DR: In this paper, the relative commutation relations between n pairs of Fermi operators and m pairs of Bose operators were defined in such a way that they generated the simple orthosymplectic Lie superalgebra B(n,m).
Abstract: It is shown that the relative commutation relations between n pairs of para‐Fermi operators and m pairs of para‐Bose operators can be defined in such a way that they generate the simple orthosymplectic Lie superalgebra B(n,m). In a case of ordinary statistics this leads to mutually anticommuting Bose and Fermi fields.

Journal ArticleDOI
TL;DR: Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered in this paper, and analysis of either derives properties of both.
Abstract: Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

Journal ArticleDOI
TL;DR: In this paper, a method of generalizing a class of invariants for a time-dependent linear oscillator is developed for the motion of a mass point in one dimension with a general timedependent nonlinear potential.
Abstract: A method of generalizing a class of invariants for a time‐dependent linear oscillator is developed for the motion of a mass point in one dimension with a general time‐dependent nonlinear potential. Formulas are derived for the allowable time‐dependent potentials and for the corresponding invariants. The method by which these conclusions are reached is interesting theoretically and is explained in detail.

Journal ArticleDOI
TL;DR: In this article, the modified fourth-order indices for Lie algebras are intimately related to eigenvalues of symmetrized fourthorder Casimir invariants, and the triality principle for the Lie algebra D4 is discussed in connection with identical vanishing of the modified 4-order index for this algebra.
Abstract: The fourth‐order indices for Lie algebras have been defined and studied by Patera, Sharp, and Winternitz. We show that it may be more convenient to modify the original definition and that the modified fourth‐order indices are intimately related to eigenvalues of symmetrized fourth‐order Casimir invariants. Explicit expressions for these quantities are given and we also find a quartic trace identity involving the generic element of these Lie algebras. We discuss the triality principle for the Lie algebra D4 in connection with identical vanishing of the modified fourth‐order index for this algebra.

Journal ArticleDOI
TL;DR: In this article, the authors consider Lagrangians s−equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential.
Abstract: Two Lagrangians are s‐equivalent (s for ‘‘solution’’) if they yield equations of motion having the same set of solutions. We consider Lagrangians s‐equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential. We show that for n=dimension of space ≥3, there are many s‐equivalent Lagrangians which cannot be formed from T−V by multiplication by a constant or addition of a total time derivative. In general these s‐equivalent Lagrangians lead to inequivalent quantum theories in the sense that the energy spectra are different.

Journal ArticleDOI
TL;DR: The formal solutions of certain three-dimensional inverse scattering problems presented in this paper were employed to obtain quantitative estimates on the error resulting from the use of the Born approximations in both direct and inverse potential scattering problems, and these estimates were uniformly valid at all energies, and for all sufficiently weak potentials.
Abstract: The formal solutions of certain three‐dimensional inverse scattering problems presented in papers I–III in this series [J. Math. Phys. 10, 1819 (1969); 17, 1175 (1976); 21, 2648 (1980)] are employed here to obtain quantitative estimates on the error resulting from the use of the Born approximations in both direct and inverse potential scattering problems. These estimates are uniformly valid at all energies, and for all sufficiently weak potentials.