Showing papers in "Journal of Mathematical Physics in 1981"

Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach

Abstract: Concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory. This makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis, but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses. Two alternative formalisms, labeled the ’’forward sensitivity formalism’’ and the ’’adjoint sensitivity formalism,’’ are developed in order to evaluate the sensitivity of the response to variations in the system parameters. The forward sensitivity formalism is formulated in normed linear spaces, and the existence of the Gâteaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity. This formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses. On the other hand, for problems involving many parameter alterations or a large data base and comparatively few functional‐type responses, the alternative adjoint sensitivity formalism is computationally more economical. However, it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism. In particular, the requirement that operators acting on the state vector and on the system parameters must admit densely defined Gâteaux derivatives is shown to be of fundamental importance for the validity of this formalism. The present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations.

Solution of Poisson’s equation: Beyond Ewald‐type methods

Abstract: A general method for solving Poisson’s equation without shape approximation for an arbitrary periodic charge distribution is presented The method is based on the concept of multipole potentials and the boundary value problem for a sphere In contrast to the usual Ewald‐type methods, this method has only absolutely and uniformly convergent reciprocal space sums, and treats all components of the charge density equivalently Applications to band structure calculations and lattice summations are also discussed

Sensitivity theory for nonlinear systems. II. Extensions to additional classes of responses

Abstract: This work extends a recent, functional‐analytic formulation of sensitivity theory to include treatment of additional types of responses. There are physical systems where a critical point of a function that depends on the system’s state vector and parameters defines the location in phase‐space where the response functional is evaluated. The Gâteaux differentials giving the sensitivities of both the functional and the critical point to changes in the system’s parameters are obtained by alternative formalisms. The foward sensitivity formalism is the simpler and more general, but may be prohibitively expensive for problems with large data bases. The adjoint sensitivity formalism, although less generally applicable and requiring several adjoint calculations, is likely to be the only practical approach. Sensitivity theory is also extended to include treatment of general operators, acting on the system’s state vector and parameters, as response. In this case, the forward sensitivity formalism is the same as for functional responses, but the adjoint sensitivity formalism is considerably different. The adjoint sensitivity formalism requires expanding the indirect effect term, an element of a Hilbert space, in terms of elements of an orthonormal basis. Since as many calculations of adjoint functions are required as there are nonzero terms in this expansion, careful consideration of truncating the expansion is needed to assess the advantages of the adjoint sensitivity formalism over the forward sensitivity formalism.

Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect

Abstract: A recent paper established technical conditions for the construction of a class of induced representations of the nonrelativistic current group SΛK, where S is Schwartz’s space of rapidly decreasing C∞ functions, and K is a group of C∞ diffeomorphisms of Rs. Bose and Fermi N‐particle systems were recovered as unitarily inequivalent induced representations of the group by lifting the action of K on an orbit Δ⊆S′ to its universal covering space δ. For s⩾3, δ is the coordinate space for N particles, which is simply connected. In two‐dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov–Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of Rs. The vector potential plays no role in this description of the Aharonov–Bohm effect.

Some models of anisotropic spheres in general relativity

Abstract: A heuristic procedure is developed to obtain interior solutions of Einstein’s equations for anisotropic matter from known solutions for isotropic matter. Five known solutions are generalized to give solutions with anisotropic sources.

Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems

Abstract: We start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates. We use Kolmogorov–Arnold–Moser (KAM) theory to show that some of the resulting tori persist under small perturbations and use a vector of Melnikov integrals to show that, under suitable hypotheses, their stable and unstable manifolds intersect transversely. This transverse intersection is ultimately responsible for Arnold diffusion on each energy surface. The method is applied to a pendulum–oscillator system.

Dimension and Character Formulas for Lie Supergroups

Abstract: A character formula is derived for Lie supergroups. The basic technique is that of symmetrization and antisymmetrization associated with Young tableaux generalized to supergroups. We rewrite the characters of the ordinary Lie groups U(N), O(N), and Sp(2N) in terms of traces in the fundamental representation. It is then shown that by simply replacing traces with supertraces the characters of certain representations for U(N/M) and OSP(N/2M) are obtained. Dimension formulas are derived by calculating the characters of a special diagonal supergroup element with (+1) and (−1) eigenvalues. Formulas for the eigenvalues of the quadratic Casimir operators are given. As applications, the decomposition of a representation into representations of subgroups is discussed. Examples are given for the Lie supergroup SU(6/4) which has physical applications as a dynamical supersymmetry in nuclei.

Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics

Abstract: After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S‐matrix poles in the lower half of the second sheet and exponentially growing vectors from the S‐matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.

The lump solutions and the Bäcklund transformation for the three‐dimensional three‐wave resonant interaction

Abstract: A Backlund transformation is found for the three‐dimensional three‐wave resonant interaction, and from it, N‐lump exact solutions may be constructed. The one‐lump solution is analyzed in detail, and it is shown that it describes such effects as pulse decay, upconversion, and explosive instabilities, all in three dimensions.

Factorization of operators.II

Abstract: We extend the methods of a previous paper [J. Math. Phys. 21, 2508 (1980)], factorizing the general, scalar, third‐order differential operator, and obtain a Miura transformation for the Boussinesq equation. We give a general factorized eigenvalue problem. We also give a Hamiltonian structure associated with the factorized eigenvalue problem. We derive several isospectral flows, some of Klein–Gordon type.

Linkages in general relativity

Abstract: For an asymptotically flat space–time in general relativity there exist certain integrals, called linkages, over cross sections of null infinity, which represent the energy, momentum, or angular momentum of the system. A new formulation of the linkages is introduced and applied. It is shown that there exists a flux, representing the contribution of gravitational and matter radiation to the linkage. A uniqueness conjecture for the linkages is formulated. The ambiguities due to the possible presence of supertranslations in asymptotic rotations are studied using the behavior of the linkages under first‐order perturbations in the metric. While in certain situations these ambiguities disappear in the first‐order treatment, an example is given which suggests that they are an essential feature of general relativity and its asymptotic structure.

Quantum field theory of singletons. The Rac

Abstract: Dirac singletons are the most elementary physical objects on de Sitter space; massless particles are two‐singleton states. Singleton field theory is characterized by a gauge structure that is strikingly similar to that of massless particles. The Rac and the Di are described by a scalar field and a spinor field, respectively, so the appearance of a fully developed gauge structure is remarkable. In this paper Rac field theory is quantized by using an indefinite metric, in close analogy with the Gupta–Bleuler method of electrodynamics. A Lagrangian theory of interacting Rac fields is also given, including the unique invariant local‐interaction between Racs.

Radiative degrees of freedom of the gravitational field in exact general relativity

Abstract: The radiative degrees of freedom of the gravitational field are isolated by analyzing the structure available at null infinity, JIt is shown thay they are coded in certain equivalence classes {D} of connections; all information about gravitational radiation can be extracted from the curvature tensors of these connections directly on J without any reference to the interior of space–time. The space of classical vacua—i.e., of {D} with trivial curvature—is analyzed. It is shown that the quotient ST/T of the BMS supertranslation group by its translation subgroup acts simply and transitively on this space. The available structure is compared with that of gauge theories. Since the entire discussion can be carried out onJ without any reference to the interior, it suggests a new approach to quantum gravity. This approach will be presented in detail in a subsequent paper.

Proof of a Geroch conjecture

Abstract: We prove that any given stationary axisymmetric vacuum space‐time (SAV) can be generated from Minkowski space by at least one Kinnersley–Chitre transformation, i.e., by at least one member of the Geroch group K, provided that the metric tensor and the Killing vectors are C3 in a domain which covers at least one point of the axis at which one of the Killing vectors characterizing the space‐time is timelike. We find that the set of all Kinnersley–Chitre transformations which map any given SAV into another given SAV is uniquely determined by the initial and final values of the Ernst potential on the axis. An explicit formula for these K‐C transformations in terms of the initial and final axis values is given; this formula generalizes an analogous one which Xanthopoulos found for the asymptotically flat SAV’s.

Representations of supergroups

Abstract: An explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations. The rules of supersymmetrization are characterized by extended Young supertableaux. This constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all Casimir operators can be calculated. We have applied this approach so far to the supergroups SU(N/M), OSP(N/2M), P(N), for which we have obtained all the representations constructible as direct products of the fundamental (defining) representations. An argument is presented toward the irreducibility of all these representations.

Geometrical structure of Faddeev–Popov fields and invariance properties of gauge theories

Abstract: Let Pn be the trivial principal bundle with structural group G and base space Pn−1, P1 being the usual fiber bundle of gauge theories. In order to give a geometrical interpretation to the Faddeev–Popov fields, as well as to the Becchi, Rouet, and Stora transformations, we need to use the fiber bundle P3. The gauge fields and the Faddeev–Popov ghost and antighost fields appear as part of certain 1‐forms defined on the base space P2. The anticommuting character of the ghost and antighost fields is essentially due to their identification with 1‐forms. The Becchi, Rouet, and Stora transformations are identified with generalized infinitesimal gauge transformations on P3 of parameters related to the ghost fields. We obtain a further invariance of the action given by a similar generalized infinitesimal gauge transformation on P3 related to the antighost fields.

Equivalent Lagrangians: Multidimensional case

Abstract: We generalize a theorem known for one‐dimensional nonsingular equivalent Lagrangians (L and ?) to the multidimensional case. In particular, we prove that the matrix Λ, which relates the left‐hand sides of the Euler–Lagrange equations obtained from L and ?, is such that the trace of all its integer powers are constants of the motion. We construct several multidimensional examples in which the elements of Λ are functions of position, velocity, and time, and prove that in some cases equivalence prevails even if detΛ = 0.

Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform

Abstract: The main purpose of this paper is to describe a technique of reduction, whereby from the class of evolution equations for matrices of order N solvable via the spectral transform associated to the (matrix) linear Schrodinger eigenvalue problem, one derives subclasses of nonlinear evolution equations involving less than N2 fields. To illustrate the method, from the equations for matrices of order 2 two subclasses of equations for 2 fields (rather than 4) are obtained. The first class coincides, or rather includes, that solvable via the spectral transform associated to the generalized Zakharov–Shabat spectral problem; further reduction to nonlinear evolution equations for a single field reproduces a number of well‐known equations, but also yields a novel one (highly nonlinear). The second class also yields highly nonlinear equations; some examples are given, including another novel evolution equation for a single field.

Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media

Abstract: For the dispersion of waves in a homogeneous medium there exist the Kramers–Kronig relations for the wave number K(ω) = ω/c(ω) The usual mathematical proof of such relations depends on assumptions for the asymptotic behavior of c(ω) at high frequency, which for electromagnetic waves in dielectrics can be evaluated from the microphysical properties of the medium In this paper such assumptions are removed and the necessary asymptotic behavior is shown to follow the representation of K(ω) as a Herglotz function From the linear, causal, and passive properties of the media we thus establish the Kramers–Kronig relations for all linear wave disturbances including acoustic, elastic, and electromagnetic waves in inhomogeneous as well as homogeneous media without any reference to the microphysical structure of the medium

Super Lie groups: global topology and local structure

Abstract: A general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous to theorems in classical Lie group theory) are proved. The relationship of the super Lie groups defined here to the formal groups of Berezin and Kac and the graded Lie groups of Kostant is analyzed.

Schrödinger equation with time‐dependent boundary conditions

Abstract: A Schrodinger equation for a well potential with varying width is studied. Generalized canonical transformations are shown to transform the problem into a time‐dependent harmonic oscillator problem submitted to fixed boundary conditions. This transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well. Motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables.

On the inverse problem of the calculus of variations

Abstract: We consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first‐order form We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach We present examples of first‐order Lagrangians for systems for which no second‐order Lagrangians exist The Hamiltonian theory for first‐order (degenerate) Lagrangians is constructed using Dirac’s method for singular Lagrangians

All nontwisting N’s with cosmological constant

Abstract: The set of all essentially different nontwisting type‐N solutions with cosmological constant is presented in such a tetrad gauge and coordinatization that the metric depends linearly on an arbitrary structural function. The special branches of the solutions of this type are shown to amount to the contractions of the most general branch.

On the existence of neutrino ’’zero‐modes’’ in vacuum spacetimes

Abstract: Neutrino ’’zero modes’’ in curved spacetime, the analog of static solutions of the neutrinoequation in flat space, are defined as the kernel of an elliptic operator obtained from a ’’3+1’’ decomposition of the neutrinoequation relative to a spacelike hypersurface. In this paper, vacuum, globally hyperbolic spacetimes that admit ’’zero modes’’ are characterized.

General relativity: Dynamics without symmetry

Abstract: The concept of conditional symmetry is introduced for a parametrized relativistic particle model and generalized to geometrodynamics. Its role in maintaining a one‐system interpretation of the quantized theory is emphasized. It is shown that geometrodynamics does not have any conditional symmetry: Such a symmetry should be generated by a dynamical variable K[gab, pab] which is linear and homogeneous in the gravitational momentum pab and which has a weakly vanishing Poisson bracket with the super‐Hamiltonian and supermomentum. The generators K fall into equivalence classes modulo the supermomentum constraint. It is shown that each equivalence class can be represented by a member which is a spatial invariant. The remaining weak equations are turned into strong equations by the method of Lagrange multipliers. The local structure of the super‐Hamiltonian and supermomentum imposes locality restrictions on the multipliers. These restrictions imply that the generator must be weakly equivalent to a local generator. A recursive argument then shows that the local generator must actually be weakly ultralocal. This uniquely determines the generator as the conformal Killing (super)vector of the local supermetric. However, the curvature scalar in the super‐Hamiltonian breaks the conditional symmetry of the supermetric term and turns geometrodynamics into a theory without any symmetry. This result is generalized to inhomogeneous generators.

An exact invariant for a class of time‐dependent anharmonic oscillators with cubic anharmonicity

Abstract: An exact invariant is constructed for a class of time‐dependent anharmonic oscillators using the method of the Lie theory of extended groups. The presence of the anharmonic term imposes a constraint on the nature of the time dependence. For a subclass it is possible to obtain an energy‐like integral and a condition under which the motion is bounded.

Universal formats for nonlinear ordinary differential systems

Abstract: It is shown that very general nonlinear ordinary differential systems (embracing all that arise in practice) may, first, be brought down to polynomial systems (where the nonlinearities occur only as polynomials in the dependent variables) by introducing suitable new variables into the original system; second, that polynomial systems are reducible to ’’Riccati systems,’’ where the nonlinearities are quadratic at most; third, that Riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity. The elemental systems have representations as novel types of matrix Riccati equations. Different starting systems and their associated Riccati systems differ from one another, at the final elemental level, in order and in initial data, but not in format.

Dual‐mass in general relativity

Abstract: A general notion of dual‐mass, the gravitational analog of the magnetic monopole, is formulated in space‐times that are asymptotically empty and flat at null infinity and in which the Bondi news vanishes. Dual‐mass is specified by a real valued linear function on the asymptotic infinitesimal translation symmetries which, furthermore, depends on the asymptotic dual Weyl curvature tensor. It is shown that space‐times with nonzero dual‐mass are characterized by a null boundary (null infinity) having the structure of a principal S1 fiber bundle over S2 such that the dual‐mass is proportional to the number of twists, n, in the bundle. Thus the topology of null infinity is that of a lens space L(n,1). A consequence of the existence of dual‐mass is that the space‐time is acausal. The NUT space‐time is shown to be an example exhibiting these features, with a null infinity having the three‐sphere topology.

Quasigroup construction and first class constraints

Abstract: A generalization of the Lie group construction is proposed wherein the composition law depends, apart from the parameters of the transformations composed, also on the transformed variables This construction is met, in particular, on the hypersurfaces specified by the first class constraints in phase space

Realization, extension, and classification of certain physically important groups and algebras

Abstract: An associative algebra of differential forms with division has been constructed. The algebra of forms in each different space provides a practical realization of the universal Clifford algebra of that space. A classification of all such algebras is given in terms of two distinct types of algebras Nk and Sk. The former include the dihedral, quaternion, and Majorana algebras; the latter include the complex, spinor, and Dirac algebras. The associative product expresses Hodge duality as multiplication by a basis element. This makes possible the realization of higher order algebras in a calculationally useful algebraic setting. The fact that the associative algebras, as well as the enveloped Lie algebras, are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws.