Volume 33, Issue 5, May 1992
Index of content:

SL(3,R) as the group of symmetry transformations for all one‐dimensional linear systems. III. Equivalent Lagrangian formalisms
View Description Hide DescriptionThe SL(3,R) theory of projective transformations of the plane is applied to the Lagrangians of all one‐dimensional Newtonian linear systems. Noether and non‐Noether equivalent Lagrangians, as well as the associated Noether and non‐Noether constants of motion, are thus obtained in a completely general and systematic way. Complete unification is achieved by this group‐theoretic approach to Lagrangians of one‐dimensional linear systems.

Real representations of Clifford algebras
View Description Hide DescriptionSimple proof of classification theorem of all real irreducible representations of Clifford algebras is given.

Racah–Wigner calculus for the super‐rotation algebra. I
View Description Hide DescriptionThe symmetry properties and the pseudo‐orthogonality relations of the super‐rotation Clebsh–Gordan coefficients for the tensor product of two irreducible representations of the super‐rotation algebra are derived. The symmetric super‐rotation 3‐j symbol and the symmetric and invariant super‐rotation 6‐j symbol are defined, their basic properties are described, and their relations to the usual 3‐j and 6‐j symbols are given.

Algebraic structure of tensor superoperators for the super‐rotation algebra. II
View Description Hide DescriptionIt is shown that the sets of tensor superoperators for the super‐rotation algebra can be used to build explicit bases for the representations of several superalgebras. The representations built in this way are the fundamental representations of the special linear superalgebras sl(2j+1‖2j) and of the orthosymplectic superalgebras osp(2j+1‖2j) and the (4j+1)‐dimensional representations of osp(1‖2) and sl(1‖2) (Stavraki) superalgebras. It is shown that the chain osp(1‖2)⊆osp(2j+1‖2j) or osp(2j‖2j+1) explains the existence of a series of nontrival zeros for the super‐rotation 6‐j symbol (SR6‐j symbols).

Methods to determine generators and defining relations for an n‐dimensional point group
View Description Hide DescriptionFour methods to determine a generating set of matrices satisfying defining relations of an n‐dimensional point group are described, given all point group elements or given a generating set, but without knowing defining relations. The first method makes use of the stepwise construction of the point group, adding a new generator at each step. Defining relations are achieved by making a coset decomposition at each step. Another approach is by use of the lower central series, yielding generators and relations according to a well‐known technique. If the point group is nonsolvable, there is a nontrivial subgroup which has itself as commutator group. The first method is then used to treat this subgroup. This approach is denoted as the second method. The third new method is a depth‐first search method. At each step in the construction, a certain element is multiplied with a certain generator according to the depth‐first search rule. In the fourth method, which is similar to the third method, the point group is constructed according to a breadth‐first search rule. The first method gives better results than the other methods with regard to (w.r.t.) the number of defining relations and the computer time.

On the topological phases in a planar N‐point vortex system
View Description Hide DescriptionA topological phase is given for the planar N‐point vortex problem. This phase arises from an element in the pure braid group, and is independent of the evolution equations. It is deduced from a regular representation of the group of diffeomorphisms on the plane. This resulting representation of the pure braid group is the monodromy of the hypergeometric functions in N variables. The relations between this topological phase and the usual geometrical phase are discussed.

Orbit–orbit branching rules between simple low‐rank algebras and equal‐rank subalgebras
View Description Hide DescriptionComplete orbit–orbit branching rules, or equivalently, reduction of Weyl group orbits, between each simple algebra of rank 3 or less, and its equal‐rank subalgebras and between F _{4} and each of its equal‐rank subalgebras are given. The generic case A _{ n }⊇A _{ n−1}×U(1), the subjoining F _{4}≳B _{3}×U(1), and E _{8}⊇E _{6}×A _{2} (first and seventh highest weight labels nonzero) are also treated. Branching rules between rank 4 and 5 simple algebras and E _{6} and their equal‐rank subalgebras are available in a depository.

Realization of the discrete series of unitary representations of Sl(2,R) in terms of creation and annihilation operators
View Description Hide DescriptionA basis for the Hilbert space of the discrete series of unitary representations of Sl(2,R) is constructed as a complete set of coherent states of pairs of particle–antiparticle creation operators. The integer q specifying an irreducible representation is the eigenvalue of the charge operator. Representations with charges ∓q are equivalent.

Graded algebras and geometry based on Yang–Baxter operators
View Description Hide DescriptionThe Γ‐graded algebras such as the ε‐symmetric algebra, generalized Lie algebras and the concept of generalized Lie–Cartan pairs are given in terms of graded Yang–Baxter operators.

Analysis on the p‐adic superspace. I. Generalized functions: Gaussian distribution
View Description Hide DescriptionThe theory of generalized functions on the p‐adic superspace Q ^{ n,m } _{ p } over the (super)commutative Banach superalgebra Λ=Λ_{0}⊕Λ_{1} with the trivial Λ_{1} annihilator is proposed. The p‐adic Gaussian ‘‘supermeasure’’ is defined as a generalized function on the p‐adic superspace. The connection between the p‐adic Gaussian integration and the p‐adic gamma function of Morita is considered

Analysis on the p‐adic superspace. II. Differential equations on p‐adic superspace
View Description Hide DescriptionThe Cauchy problem for differential equations on the p‐adic superspace is considered. The application of this mathematical theory to the model of the supersymmetric quantum mechanics on the p‐adic Riemannian surface is proposed. The non‐Archimedean superdiffusion is also considered.

Functional calculus of the Borel transform
View Description Hide DescriptionAn explicit form is given of the Borel transform B̃ of g■f where g■f(z):=g ( f ( z )) with f Borel summable and g analytic in f(0) and a detailed study of the singularities in B̃ (in the Borel variable) is made. Examples include g(ξ)=1/ξ, log ξ and e ^{ξ}. The present paper generalizes results of Boenkost et al. [J. Math. Phys. 29, 1118 (1987)].

Jet methods in nonholonomic mechanics
View Description Hide DescriptionClassical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle τ: M→R. The framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly. For systems with linear constraints a new interpretation is given of Appell’s equations as first‐order differential equations associated with a suitable vector field. The d’Alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints. The equivalence between the equations of motion arising from this generalization, the ones set up on Gauss’ principle of least constraint and Hertz’s principle of least curvature is established.

A new asymptotic form for the Laguerre polynomials
View Description Hide DescriptionA new asymptotic formula in terms of elementary functions is demonstrated for the Laguerre polynomialsL ^{α} _{ n }(x) in the limit when all three parameters (n,α,x) become large. This asymptotic limit appears in the problem of obtaining the low‐field semiclassical approximation from the full quantum mechanical behavior of a two‐dimensional electron gas in a magnetic field.

Variational density of electrons for large closed shell atoms
View Description Hide DescriptionAn exact expression for the atomic form factor for closed shell atoms is derived using an antisymmetric multielectron wave function. It is shown that for large atoms it can be written as an integral over a Bessel function. Using the Hankel transform it gives an expression for the density of electrons which is of Thomas–Fermi form which was found earlier by Pucci and March.

Note on the representation of many‐particle dynamics as higher‐order single‐particle dynamics, with a means to relativistic elevation of Newtonian dynamics
View Description Hide DescriptionIt is shown how (first in Newtonian physics), by processes of repeated differentiation of equations of motion and of algebraic elimination, the dynamics of many particles may be brought to another equivalent representation, that of a single higher‐order equation of motion for a single particle. Here a higher‐order Lagrangian followed by an Ostrogradsky Hamiltonian may be brought into play. (This leads parenthetically to the construction of a considerable class of canonically inequivalent specifications of one basic set of equations of motion.) The higher‐order one‐particle representation translates simply and directly (but not uniquely) into relativistic generalization (because only a single world line is being described), in which the Poincaré group is canonically represented in an Ostrogradsky Hamiltonian formulation. The examples of the Kepler problem and the harmonic oscillator are elaborated in detail.

The extension theory and the opening in semitransparent surface
View Description Hide DescriptionThe model of zero width slits based on the operator’s extension theory for the wave scattering by a semitransparent surface with small aperture is constructed.

Super Wigner function
View Description Hide DescriptionThe phase‐space representation of supersymmetric quantum mechanics is given. The super Wigner function is constructed and its transformation property is investigated. In the case of a supersymmetric oscillator, it is explicitly shown how the invariance under the supersymmetry transformation determines the form of the super Wigner function.

Stochastic lengths
View Description Hide DescriptionThe autocorrelations of the line element for the sample paths of diffusion processes are computed. Lengths are defined for the sample paths of diffusions which depend upon an arbitrary function f and reduce to the classical relativistic length for differentiable trajectories. Their mean values are computed for a general diffusion and provide a family of stochastic actions labeled by f. In a variational principle, these actions are extremized over the drift and probability density and the equations characterizing the critical diffusions are derived. The Klein–Gordon equation is obtained from the variation of arbitrary f‐dependent stochastic extensions of the length of trajectories.

Novel solution of the one‐dimensional quantum‐mechanical harmonic oscillator
View Description Hide DescriptionThe classical solution is used as a variable for the one‐dimensional quantum‐mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time‐independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a δ‐function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac δ functions.