Volume 33, Issue 10, October 1992
Index of content:

A contribution to nonstandard superanalysis
View Description Hide DescriptionThe nonstandard (infinitesimal) analysis developed by Robinson is applied to certain foundational aspects of superanalysis. The construction of nonstandard hull due to Luxemburg is used to study ‘‘limit phenomena’’ arising in a Grassmann algebra Λ(q) as the number of generators q tends to infinity. An outcome of such an approach is that under transition t→∞ a superspace of purely odd dimension (0,q) transforms into a superspace of dimension (∞,∞) and not (0,∞) as one would expect. It is suggested that this construction may have something to do with the problem of adequate mathematical description of collective bosonic effects arising within infinite fermionic dimensions.

Generalized graph theory on Lie g and applications in conformal field theory
View Description Hide DescriptionEach magic basis of Lie g supports a generalized graph theory on Lie g, whose axioms include the graphs of order n as a special case on SO(n). Other examples include the sine‐area graphs of SU(n) and the sine(⊕area) graphs of SU(Π_{ in } _{ i }). The set of all (magic basis–generalized graph theory) pairs is not known, but each possible pair corresponds to a large set of new conformal field theories.

Explicit orthogonalization of some biorthogonal bases for SU(n)⊇SO(n) and Sp(4)⊇U(2)
View Description Hide DescriptionMutually related explicit algebraic‐polynomial expressions of the orthogonalization coefficients are proposed for the biorthogonal [polynomial (Bargmann–Moshinsky), stretched, antistretched, quasistretched, and their dual] bases of the two parametric (mixed tensor and covariant) irreducible representations (irreps) of SU(n) restricted to SO(n), as well as for the projected (Smirnov–Tolstoy) and dual bases of the five‐dimensional quasispin. The orthogonalization coefficients of the essentially simplified Gram–Schmidt process are expressed, up to explicitly given elementary factors, in terms of the numerator and denominator polynomials, represented as compositions of the generalized hypergeometric coefficients <_{3}F_{2}(...)‖μ≳ and <_{1}F_{0}(...)‖ν≳ and equivalent in the diagonal (denominator) and boundary cases to the A _{λ}(_{ c } ^{ abde }) functions of Biedenharn and Louck. The distribution of zeros and the symmetry properties of the introduced polynomials are crucial for the conjectured solutions.

Bicovariant differential calculus on the two‐parameter quantum group GL_{ p,q }(2)
View Description Hide DescriptionThe quantum Lie algebra and quantum de Rham complex of the two‐parameter quantum group GL_{ p,q }(2) are presented. It is shown that the differential calculus on the two‐parameter quantum group GL_{ p,q }(2) given in this paper is bicovariant.

Analytical solutions to classes of linear oscillator equations with time varying frequencies
View Description Hide DescriptionAnalytical solutions to differential equations in the form of ÿ+[q(t)+q _{1}(t)]y=0 are presented, where q _{1}(t) is dependent on the solution of the basic equationÿ+q(t)y=0. Various forms of the first equation are generated through a suggested procedure and their solutions should have some value in the mathematical physics field as well as in engineering applications.

Riemannian twistors and Hermitian structures on low‐dimensional space forms
View Description Hide DescriptionA Riemannian Kerr Theorem is described, which associates to an integrable Hermitian structure (with singularities) on S ^{4} a complex analytic surface in C P ^{3}. To such a Hermitian structure can be associated a harmonic morphism defined on each of the four‐dimensional space forms with values in a Riemann surface. This unifies the classifications for the three‐dimensional space forms of Baird and Wood and the recent classifications for R ^{4} and S ^{4} by Wood, into one simple equation. As a consequence we derive the classification for harmonic morphisms defined on four‐dimensional hyperbolic space with values in a Riemann surface and a new implicit form for harmonic morphisms defined on three‐dimensional hyperbolic space. Unification of the Kerr Theorems for S ^{4} and for compactified Minkowski space M is achieved at the complex level after suitably embedding the spaces in compactified, complexified Minkowski space.

Linearization of polynomials
View Description Hide DescriptionIt is well known that the linearization of quadratic forms is accomplished using Dirac matrices. The general problem of linearization of any polynomial of degree n having p variables is considered. First the homogeneous polynomials are considered and it is shown that we only need to study two basic homogeneous forms, namely, the sum and the product one. The sum is linearized using matrices which turn out to be a matrix representation of a generalized Clifford algebra. The homogeneous form is linearized using matrices, the size of which is large for practical use. Some clues are given to reduce their size. Since any polynomial of degree n can be made homogeneous by introducing a supplementary variable, the method proposed is quite general. It constitutes an algorithm for the linearization of any polynomial.

Stability of Hamiltonian systems at high energy
View Description Hide DescriptionLet H=T(P)+U(P,φ) be the Hamiltonian for a classical mechanical system of l degrees of freedom, where T is an integrable Hamiltonian generating quasiperiodic motion in the first l−1 degrees of freedom which has the form

Momentum operators with gauge potentials, local quantization of magnetic flux, and representation of canonical commutation relations
View Description Hide DescriptionCommutation properties of two‐dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrödinger representation of the canonical commutation relations.

Moyal quantization of 2+1‐dimensional Galilean systems
View Description Hide DescriptionStratonovich–Weyl kernels are constructed for some of the coadjoint orbits of the two‐dimensional extended Galilean group G(2+1). As an intermediate step, the unitary irreducible representations associated with a given group orbit are obtained by using the Kirillov–Mackey theory. Star products are defined, in the sense of Moyal, for functions on each of these orbits. The central extension of G(2+1) with parameter k is also analyzed, which results from the commutator between the generators of boosts, to conclude that it originates a sort of nonrelativistic remainder of the Thomas precession.

On pararelativistic quantum oscillators
View Description Hide DescriptionDifferent choices of matrices characterizing p=2 parafermions are analyzed in connection with the description of relativistic spin‐one particles through the Kemmer formulation. The free and interacting cases are considered and the relations between parasupersymmetry and Kemmer theory are enhanced as it is also the case between supersymmetry and Diractheory. In that way the oscillatorlike context leads to the characterization of pararelativistic oscillators.

Rotational diffusions as seen by relativistic observers
View Description Hide DescriptionThe major unsolved problem in the framework of Nelson’s stochastic mechanics is addressed and an attempt is made to provide a description of relativistic spin‐1/2 particles in terms of Markovian diffusions on S _{3}. Random rotations are here labeled by the proper time of a particle in relativistic motion and are continuously distributed along a space‐time trajectory followed by the particle in Minkowski space.

Green’s function for crossed time‐dependent electric and magnetic fields. Phase‐space quantum mechanics approach
View Description Hide DescriptionIn the present paper, the Moyal propagator is obtained for an electron inmersed in a constant uniform magnetic field crossed with a uniform electric field having arbitrary time dependence, using the techniques of phase space quantum mechanics. All the interesting magnitudes can be evaluated from it, and one can make quantum physics remaining in the context of phase space. Making use of these methods, the corresponding Green’s function is also obtained.

Algebraic treatment of a general noncentral potential
View Description Hide DescriptionIt is shown that the problem of a particle moving in a noncentral potential generalizing the Coulomb potential and the Hartmann ring‐shaped potential accepts SO(2,1)⊗SO(2,1) as a dynamical group, within the framework of the Kustaanheimo–Stiefel transformation. The Green’s function relative to this compound potential is calculated through the algebraic approach so(2,1) and with the help of the Baker–Campbell–Hausdorff formulas. The energy spectrum and the normalized wave functions of the bound states can then be deduced. Eventually, the Coulomb potential, the Hartmann ring‐shaped potential and also, due to its close link with the latter, the compound Coulomb plus Aharonov–Bohm potential may all be considered as particular cases.

Duality for the matrix quantum group GL_{ p,q }(2,C)
View Description Hide DescriptionIt is shown that the algebraU_{ p,q } dual to GL_{ p,q }(2,C) is isomorphic to U_{(pq)} ^{1/2} (sl(2,C)) ⊗ Z as a commutation algebra, where Z is a subalgebra central in U_{ p,q }. The subalgebra Z is a Hopf subalgebra of U_{ p,q }, while the commutation subalgebra U_{(pq)} ^{1/2} (sl(2,C)) is not a Hopf subalgebra.

C‐statistical quantum groups and Weyl algebras
View Description Hide DescriptionC‐statistical quantum groups and algebras are obtained by a process of transmutation. They are like super‐quantum groups and algebras but with statistics given now by a complex number. Among the examples, it is shown that the Weyl algebra of canonical quantization may be viewed equivalently as a C‐statistical plane. Related examples are the ‘‘noncommutative torus’’ and a double loop‐variable quantization of photons. These results lead toward a reformulation of ordinary quantum mechanics as a classical theory with C‐statistics. In this reformulation, the role of the ±1 factors of super statistics is played by the free‐particle time evolution.

Boson–fermion and baryon mapping: Construction of collective subspaces. I. Theory
View Description Hide DescriptionRecently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson–fermion and the baryon mapping. In the present paper, the case of a restriction to a subset of three‐fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson–fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three‐fermion subsystems carry collective indices. Examples of simple applications are presented in the two subsequent papers of this series.

Boson–fermion and baryon mapping: Construction of collective subspaces. II. Example: A simple quark shell model
View Description Hide DescriptionThe formalism derived for the boson–fermion and the baryon mapping in Paper I of this series is applied to the single‐orbit quarkshell model developed by Petry and co‐workers for the description of nuclear properties. The multiquark space is reduced to the collective subspace of multinucleon states. A collective representation of states and operators in terms of bosons and ideal fermions or in terms of ideal baryons, respectively, is constructed. The results of the original model in the multinucleon space are exactly reproduced. Finally, the two transformations are compared with a related mapping technique recently published by Pittel, Engel, Dukelsky, and Ring.

Boson–fermion and baryon mapping: Construction of collective subspaces. III. Application of the baryon mapping to many‐baryon states
View Description Hide DescriptionThe theory developed in Paper I of this series is applied to the general case of a system consisting of colorfree quark triplets, whose quantum numbers are chosen as collective trifermion indices. The appropriate mapping technique to be used here is the baryon mapping. It is demonstrated that the original multiquark states can be exactly represented by states of colorfree ideal baryons. Besides, collective extended images are derived for a class of fermionic operators leaving the collective fermion space invariant.

Evaluation of Schwinger–de Witt kernels for spin‐_{2̄} ^{1} fields through the SO(2,1) Lie algebra
View Description Hide DescriptionUsing the noncompact SO(2,1)∼SU(1,1) Lie algebra the Schwinger–de Witt kernels for massive spin‐1/2 fields are obtained in some classical background gravitational fields. The method makes use of the Schwinger–de Witt proper time representation and Baker–Campbell–Hausdorff formulas.