Volume 26, Issue 11, November 1985
Index of content:

Boson representations of the real symplectic group and their applications to the nuclear collective model
View Description Hide DescriptionBoth non‐Hermitian Dyson and Hermitian Holstein–Primakoff representations of the Sp(2d,R) algebra are obtained when the latter is restricted to a positive discrete series irreducible representation 〈λ_{ d } +n/2,...,λ_{1}+n/2〉. For such purposes, some results for boson representations, recently deduced from a study of the Sp(2d,R) partially coherent states, are combined with some standard techniques of boson expansion theories. The introduction of Usui operators enables the establishment of useful relations between the various boson representations. Two Dyson representations of the Sp(2d,R) algebra are obtained in compact form in terms of ν=d(d+1)/2 pairs of boson creation and annihilation operators, and of an extra U(d) spin, characterized by the irreducible representation [λ_{1}⋅⋅⋅λ_{ d }]. In contrast to what happens when λ_{1}=⋅⋅⋅=λ_{ d }=λ, it is shown that the Holstein–Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation. Explicit expansions are, however, obtained by extending the Marumori, Yamamura, and Tokunaga method of boson expansion theories. The Holstein–Primakoff representation is then used to prove that, when restricted to the Sp(2d,R) irreducible representation 〈λ_{ d } +n/2,...,λ_{1}+n/2〉, the d n‐dimensional harmonic oscillator Hamiltonian has a U(ν)×SU(d) symmetry group. Finally, the results are applied to the Sp(6,R) nuclear collective model to demonstrate the existence of a hidden U(6)×SU(3) symmetry in this model.

An inverse problem in crystallography
View Description Hide DescriptionCoordinate‐independent formulations are derived for the conditions satisfied by the elastic modulustensor of materials with trigonal, hexagonal, and cubic symmetry. Analogous results for two‐dimensional modulustensors are also derived.

On combinatoric determination of SU(n) weight multiplicities
View Description Hide DescriptionA combinatoric method for evaluating the inner multiplicity of a central weight is given using results from the theory of distributions. A formula for special cases is derived, and a recursive algorithm is presented for the general problem.

On the zeros of eigenfunctions of polynomial differential operators
View Description Hide DescriptionIt is shown that for polynomial eigenfunctions of an ordinary polynomial differential operator with coefficients depending only on the independent variable it is possible to determine the density of nodes around the mean without solving the corresponding eigenvalue problem. This is done by means of the first few moments, which can be directly expressed in terms of the above‐mentioned coefficients. Also, very simple expressions for the asymptotic values (i.e., when the degree of the polynomial becomes very large) of these quantities are found. For illustration, these results are applied to various orthogonal polynomials, which satisfy ordinary differential equations of second, fourth, and/or sixth order.

Solution of an extremum problem pertaining to analytic extrapolation techniques
View Description Hide DescriptionA preliminary problem appearing in analytic extrapolation procedures when the ‘‘physical data’’ are given on a line contained in the analyticity domain and their errors are measured through the norm of the supremum is solved. As an interesting by‐product, Blaschke functions are constructed, which, in some sense, generalize the usual Chebyshev polynomials.

Classical SU(2) Yang–Mills–Higgs system: Time‐dependent solutions by similarity method
View Description Hide DescriptionA similarity analysis of the Wu–Yang–’t Hooft–Julia–Zee‐ansatz‐reduced system of nonlinear differential equations of classical SU(2) Yang–Mills–Higgs theory is presented. This yields the similarity group G of the equations. Considering G and one of its subgroups denoted G_{1}, some previously known time‐dependent solutions in the Prasad–Sommerfield limit are generated. Two new time‐dependent solutions are also reported.

On the existence of global integral forms on supermanifolds
View Description Hide DescriptionIt is shown that the Berezin approach to integration on supermanifolds can be applied to cases where the supermanifold is a twisted extension of a real manifold. This is done by showing that supermanifolds admit a subatlas of coordinate charts with transition functions of a quite restricted kind.

Mathematical aspects of quantum fluids. I. Generalized two‐cycles of ^{4}He type
View Description Hide DescriptionIt is shown that the two‐cocycle involved in the Hamiltonian description of the superfluid^{4}He, both nonrotating and rotating, is a particular case of generalized symplectic two‐cocycles on semidirect product Lie algebras.

On a class of differential equations derived from plasma statistics
View Description Hide DescriptionA class of second‐order differential equations stemming from an equation of motion for the two‐particle spatialcorrelation function in one‐component plasmas is studied. These equations contain an irregular singularity of varying order at the origin. The general form of solution is obtained which, together with the construction of asymptotic series, demonstrates that both solutions to all equations in the class are singular at the origin. Behavior removed from the origin is shown to be oscillatory or exponential depending on specifics of the equations.

Reciprocal relations for effective conductivities of anisotropic media
View Description Hide DescriptionAny pair of two‐dimensional anisotropic media with local conductivity tensors that are functions of position and that are related to one another in a certain reciprocal way are considered. It is proved that their effective conductivity tensors are related to each other in the same way for both spatially periodic media and statistically stationary random media. An inequality involving the effective conductivity tensors of two three‐dimensional media that are reciprocally related is also proved. These results extend the corresponding results for locally isotropic media obtained by Keller, Mendelsohn, Hansen, Schulgasser, and Kohler and Papanicolau. They also yield a relation satisfied by the effective conductivity tensor of a medium reciprocal to a translated or rotated copy of itself.

The structures of generalized noncommutative Toda lattices
View Description Hide DescriptionNon‐Abelian Toda lattices with a finite number of ‘‘above‐diagonal’’ variables are related to appropriate Lie algebras of operators and a certain two‐cocycle.

Formulation of Noether’s theorem for Fokker‐type variational principles
View Description Hide DescriptionA formulation of Noether’s theorem is given for Fokker‐type variational principles describing directly interacting particles. Many‐body as well as two‐body interactions depending at most on the particle positions and velocities are considered. Invariance up to a divergence of the action integral under infinitesimal transformations, as usual, leads to divergences that equal linear combinations of the Lagrangian derivatives. Conservation laws can be obtained when the Lagrangian derivatives vanish. The use of the formulation, which is independent of any specific transformations, is illustrated by rederiving the form of the conserved quantities following from the invariance of general two‐body Fokker‐type variational principles under the infinitesimal transformations of the Lorentz group and of the Galilei group; such conservation laws were previously derived using a method that, although exploiting the symmetries of the action integral, did not directly connect the divergence of the conserved quantities with linear combinations of the Lagrangian derivatives. Other applications and extensions are discussed.

On nonlocal point interactions in one, two, and three dimensions
View Description Hide DescriptionThree characterizations of all self‐adjoint extensions of the Laplacian in one, two, and three dimensions are discussed.

Uncertainty relations in stochastic mechanics
View Description Hide DescriptionPosition–momentum uncertainty in Nelson’s stochastic mechanics [Phys. Rev. 1 5 0, 1079 (1966)] has previously been investigated by de la Peña‐Auerbach and Cetto [Phys. Lett. A 3 9, 65 (1972)]. In this paper their result is generalized, and full equivalence between the uncertainty relations in stochastic mechanics and conventional quantum mechanics is established. Force–momentum uncertainty is also considered.

Solution of the Schrödinger equation for a particle in an equilateral triangle
View Description Hide DescriptionThe complete solution for the quantum‐mechanical problem of a particle in an equilateral triangle is derived. By use of projection operators, eigenfunctions belonging explicitly to each of the irreducible representations of the symmetry group C _{3V } are constructed. The most natural definition of the quantum numbers p and q includes not only integers but also nonintegers of the class (1)/(3) and (2)/(3) modulo 1. Some relevant features relating to symmetry and degeneracy are also discussed.

Vector coherent state representation theory
View Description Hide DescriptionA vector coherent statetheory is formulated as a natural extension of standard coherent statetheory. It is shown that the Godement representations and the coherent state representations of the Sp(N,R) groups of Rowe and of Deenen and Quesne are special cases of this more general theory.

On the spectrum of a two‐level system
View Description Hide DescriptionDoubly degenerate energy levels of the two level atom interacting with a single mode of the electromagnetic field are exactly calculated. The dependence of the number of such levels on the values of the level separation energy and a coupling constant is determined. Some general conclusions about the spectrum are drawn.

Spectral sum rule for time delay in R^{2}
View Description Hide DescriptionA local spectral sum rule for nonrelativistic scattering in two dimensions is derived for the potential class v∈L ^{4} ^{/} ^{3} (R^{2}). The sum rule relates the integral over all scattering energies of the trace of the time‐delay operator for a finite region Σ⊆R^{2} to the contributions in Σ of the pure point and singularly continuous spectra.

Three‐dimensional inverse scattering: Plasma and variable velocity wave equations
View Description Hide DescriptionExact equations governing three‐dimensional time‐domain inverse scattering are derived for the plasma waveequation and the variable velocity classical wave equation. This derivation was announced for the plasma waveequation in a short note by the authors. That work was motivated by Newton’s three‐dimensional generalization of Marchenko’s equation. This paper gives the details of the new derivation and extends it to the classical wave equation. For the time domain derivation in this paper, the scattering region is assumed to have compact support and smoothly joins the surrounding three‐dimensional infinite medium. The derivation contains the following ingredients: (1) a representation of the solution at a point in terms of its values on a large sphere, (2) the far‐field form of the Green’s function, (3) causality, and (4) information carried in the wave front of the solution. The derivation of the classical waveinverse scatteringequation requires that the velocity in the scattering region be less than that of the surrounding medium. This condition is natural, for example, in the scalar wave model of electromagneticscattering from dielectric nonconducting bodies in free space. Finally, an experiment to verify the inverse scatteringequations is proposed.

Generating infinitesimal transformations with second‐order‐infinitesimal accuracy for proving covariance of commutation relations under finite transformations
View Description Hide DescriptionAs dynamical quantization of Einstein’s gravitational theory meets unsolved problems, it is worth considering the alternative method of quantization suggested by Fermi’s quantization of special‐relativistic electrodynamics, which for that theory has been the starting point of most modern applications of quantum electrodynamics. This method avoids first‐class constraints by an alteration of the Lagrangian. In physical formulas, this introduces unwanted terms, that are at the end equated weakly to zero by auxiliary conditions. By absence of first‐class constraints, this theory could be quantized canonically, if it would not contain fermion fields. As shown by Dirac, in the presence of fermion fields the canonical commutation relations of the altered theory have to be replaced by modified commutation relations. The ultimate purpose of this and following papers is to prove the covariance of Dirac’s modified commutation relations, first under infinitesimal transformations, and thence under finite transformations. As usual, this requires the proof of existence of a conserved and invariant generator for the transformations admitted, here coordinate transformations and local Lorentz transformations of the tetrad field. For infinitesimal transformations, the conventional method of deriving from the Lagrangian density L a generator T _{1} linear in the infinitesimal parameters that determine the transformations is used. However, for guaranteeing integrability of the procedure for generating transformations of the field variables, from infinitesimal transformations to finite transformations, it is necessary to show the existence of a more accurate generator T, no longer linear in the parameters, which will by e ^{ i T } F e ^{−i T }−F generate the transformations δ̄F of the field variables F with second‐order accuracy.
While it is left to a following paper to discuss the exact form of Dirac’s modified commutation relations and to prove the conservation and covariance of the generator T and to prove that the conventional first‐order generator T _{1} will, this time b y D i r a c’s m o d i f i e d c o m m u t a t i o n r e l a t i o n s, generate the first‐order substantial variations δ̄_{1} F of the field variables, in the present paper formulas are derived for expressing the second‐order‐infinitesimal terms in the transformation formulas by means of the first‐order terms, and a method is derived for obtaining from T _{1} the more accurate generator T. Finally, it is verified that the transformations F(x)→F’(x’) =F(x)+δ̄F generated by the generators T do satisfy the transformation group property with second‐order accuracy.