Volume 25, Issue 3, March 1984
Index of content:

On the anomaly number of the classical groups
View Description Hide DescriptionThe Adler–Bell–Jackiw (ABJ) anomalies for all representations of the classical groups are calculated with the aid of a technique proposed recently for the representation theory of these groups. The method presented here is useful also for the calculations of the eigenvalues of Casimir invariants with rank higher than 3.

Algebras with anticommuting basal elements, space‐time symmetries, and quantum theory
View Description Hide DescriptionA collection A of algebras with anticommuting basal elements is investigated. It is shown that the collection A includes the quaternions, the octonions, the ‘‘algebra of color,’’ as well as other algebras familiar to the physicist. Each algebraA is a quadratic, Jordan‐admissible algebra and possesses a norm that is a generalization of the Minkowski metric. Using a Cayley–Dickson‐like process, each such algebraA in A can be embedded into a larger algebraÂ that is also in the collection A. These algebras should provide candidates for models to describe observables, color, and other phenomena encountered in particle physics.

Double‐Gel’fand boson polynomials and the permutation group
View Description Hide DescriptionDouble Gel’fand polynomials of boson operators spanning the irreducible representation [m] of U(n) in U(n)*U(n) have been obtained using symmetrized linear combinations of Wigner operators of the permutation group. The normalized coefficients which occur in the polynomial representation have been expressed as linear combinations of the Young orthogonal representation matrix elements.

Infinitesimal operators and structure of the representations of the groups SO*(2n) and SO(2n) in a U(n) basis and of the groups SU*(2n) and SU(2n) in an Sp(n) basis
View Description Hide DescriptionThe infinitesimal operators of the most degenerate representations of the groups SO*(2n) and SU*(2n) are found in a discrete basis. The structure (composition series) of these representations is studied. The classification of unitary irreducible representations of these groups which belong to most degenerate series is given. The infinitesimal operators of irredicuble unitary representations of SO(2n) in a U(n) basis and of SU(2n) in an Sp(n) basis are found for the cases of highest weights (M, M, 0, ..., 0), M≥0.

Dyson representation of SU(3) in terms of five boson operators
View Description Hide DescriptionA representation of SU(3) in terms of five boson operators is proposed. It is a generalization of the Dyson–Maleev type representation used in nuclear physics with two boson operators related to the integers that label the irreducible representation of SU(3).

Charge operators in simple Lie groups
View Description Hide DescriptionCharge operators for representations of dimension less than or equal to 16 are computed in all simple Lie groups. The representations for which the charge operator reproduces the charge spectrum of leptons and quarks of one family are analyzed from a GUT point of view.

Cohomology of Lie algebras with a nontrivial center
View Description Hide DescriptionLet g be a Lie algebra and assume X∈ Z(g) (the center of g). Let φ: g→End V be such that φ(X)=1_{ V }. We show that H ^{ n } _{φ}(g, V)=0 for all n>0. The result applies to the nonrelativistic Poincaré, harmonic oscillator and Heisenberg algebras, and also to g⊕u(1) where g is a semisimple Lie algebra. We also give here the cohomology groups for the first three algebras with the adjoint action, giving explicit computations of H ^{1} and H ^{2}, respectively, for the first two algebras.

Specializations of integrable systems and affine Lie algebras
View Description Hide DescriptionWe analyze a number of restrictions on the evolution systems associated with the zero‐curvature equations corresponding to the extended Dynkin diagram A ^{(1)} _{ n−1}. The resulting specialized evolution systems contain exponential terms, like a third‐order differential equation previously derived in Ref. 5.

Supermanifolds and automorphisms of super Lie groups
View Description Hide DescriptionThe geometrical theory of supermanifolds is developed and applied to the super Lie groups. The resulting structural equations and the adjoint representation are studied and used to find the automorphisms of super Lie groups. The N=1 supergravity symmetry group (the so‐called graded Poincaré group) is explicitly studied and possible applications are outlined.

Finite quantum processes
View Description Hide DescriptionA model for quantum dynamics is presented in terms of a generalization of Markov chains. We first consider measurements and events on an amplitude space. Markov amplitude chains on an amplitude space are then studied. Quantum chains are defined and characterized by a Markov and weak stationarity property. We then consider random phase transformations and study the changes that result in an amplitude space and in the quantum chains due to such transformations. A perturbation expansion involving a potential perturbing the ‘‘free’’ motion is proved. Quantum processes are defined and their relation to quantum chains is discussed. The existence of an arbitrary quantum chain is proved using path chains.

Continued fraction expansions for the complete, incomplete, and relativistic plasma dispersion functions
View Description Hide DescriptionIn our investigation of the linear theory of waves in plasma and the stability of relativistic beam‐plasma systems, we have been led to consider methods for the evaluation of integrals of the form ∫dχ(χ−ζ)^{−} ^{1} exp(−χ^{2}) and ∫dχ(χ−ζ)^{−} ^{1} exp[(1−χ^{2})^{−} ^{1} ^{/} ^{2}] for complex ζ. In this work, we report on the evaluation of this integral and its derivative by means of continued fraction expansions. The expressions derived allow the precise calculation of these integrals in previously inaccessible regions. Additionally, applications to non‐Maxwellian particle distributions, such as those found in the analysis of plasma diodes, are included.

Scattering by a penetrable body
View Description Hide DescriptionAn integral equation equivalent to the interface problem is derived. A numerical scheme for its solution is given. Convergence of the scheme is established.

Three‐term recursion relations for hydrogen wave functions: Exact calculations and semiclassical approximations
View Description Hide DescriptionThree‐term recursion relations with respect to the angular momentuml are given for the normalized hydrogen wave functions associated with r, p _{ r }, p (r is the radial polar coordinate in configuration representation, p _{ r } is the momentum conjugated to r, p is the radial polar coordinate in momentum representation). These three‐term recursion relations [Eqs. (4), (6), (16)] are found numerically stable in the order of decreasing l values, even for large quantum numbers. The three‐term recursion relations in r and p are used to derive semiclassical approximations for the radial wave functionsP _{ n,l }( p) and R _{ n,l }(r). These semiclassical approximations [Eqs. (67) and (84)] are valid even at the classical turning points and are still markedly good at small quantum numbers.

Explicit integrability for Hamiltonian systems and the Painlevé conjecture
View Description Hide DescriptionWe analyze a class of Hamiltonian systems in two dimensions for which we proved that a second constant of motion exists. It is shown that, using the two first integrals, the equations of motion can be written in a form which allows their integration by quadratures. An analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex‐time plane. It is shown explicitly that the systems belonging to this class possess the ‘‘weak Painlevé’’ property, i.e., their solutions in complex time can present singularities of a specific algebraic type.

First integrals for some nonlinear time‐dependent Hamiltonian systems
View Description Hide DescriptionAn observation of a simple property of canonical transformations leads to a procedure for determining first integrals for classes of Hamiltonians. In illustrative examples the most general result presently known from other methods is recovered, a new result presented, and a generalization to more than one degree of freedom discussed.

Generalized variational principles and nondifferentiable potentials in analytical mechanics
View Description Hide DescriptionThe present paper deals with variational principles in terms of hemivariational inequalities and with multivalued differential equations which are called differential inclusions in analytical mechanics. Such inequalities and inclusions are received when no restrictions of differentiability are considered.

The direct correlation function for the hard‐core potentials
View Description Hide DescriptionThe Ornstein–Zernicke relation is examined for the class of intermolecular potentials with a hard‐core term. The nonlinear integral equation relating the direct correlation function outside the hard‐core region with the inside one is proposed. A class of analytical solutions is obtained.

Lightlike contractions on Minkowski space‐time
View Description Hide DescriptionThe lightlike contractions of electromagnetic fields are discussed. In particular Dirac’s delta type behavior on null hyperplanes is established and the lightlike limit of the electromagnetic field of a spinning, charged particle is obtained.

Quantization, symmetry, and natural polarization
View Description Hide DescriptionWe discuss the notion of polarization, as defined in a geometric quantization scheme recently introduced, in terms of the role played by the evolution operator of the quantum system. The analysis uses an integral transform representation of the group WSp(2,R). This clarifies the group theoretic origin of the natural polarizations and the meaning of the polarization changing transformations.

Orthogonality and orthocomplementations in the axiomatic approach to quantum mechanics: Remarks about some critiques
View Description Hide DescriptionThe logic approach to axiomatic quantum mechanics via orthocomplemented partial ordered sets of yes–no measurements, which constitute the observing part of a concretely realizable experiment on microworld, has been criticized from the empirical point of view by Mielnik, which on the contrary privileges the convex scheme linked to the preparing part. In this work we do assume that a description of quantum phenomenology must take into account both these two parts in which every elementary experiment can be decomposed. According to this predecision, we develop an axiomatic approach based on indistinguishability principles of a quantum information system. The very general concept of yes–no measurement or ‘‘question’’ is accepted, and then the set of all questions is classified according to the behavior with respect to a phenomenological orthogonality relation. In particular, we single out the set F of fuzzy events or effects and the set E⊆F of exact events. The Mielnik critique is then refused since it regards the order structure of E using counterexamples which pertain to F/E. The notions of physical property and noperty are then introduced and an axiomatic foundation of quantum mechanics based on a pre‐Hilbert space is discussed.