Index of content:
Volume 26, Issue 6, June 1985

On the commutant of an irreducible set of operators in real Hilbert space
View Description Hide DescriptionThe relation between irreducibility and the structure of the commutant is studied for a set of linear bounded operators on a real Hilbert space of arbitrary dimension. The results are applied to the investigation of irreducible sets of semilinear operators on a complex or quaternionic Hilbert space.

Internal structure of fermions
View Description Hide DescriptionWe investigate the equations Pψ=kψ+h with the range of ψ contained in an appropriate Clifford algebra. Next we define, on the bundle ΨM of the minimal left ideals of the Clifford algebra, the unique connection ∇̃ which produces the Christoffel connection ∇ on M. To unify the whole picture we have to introduce an additional degree of freedom α which numerates the set of global fields of mutually annihilating primitive idempotents on M. Now the geometrical ‘‘duality’ of the spinor fields given by the primary as well as by the composed character of the sections of ΨM implies the existence of some internal gauge interaction B _{μ} which changes α. The generators of the holonomy group of ∇ that do not belong to the Crumeyrolle algebra are responsible for this internal interaction.

Topological properties of unbounded bicommutants
View Description Hide DescriptionWe consider different possible definitions of unbounded commutants and unbounded bicommutants of a set or an algebra of unbounded operators. We investigate their behavior with respect to various topologies. In particular we give sufficient conditions in order that bicommutants be the closure of the original set of operators with respect to some of those topologies. We investigate some special classes of algebras (symmetric, self‐adjoint, regular, V* algebras) for which several or all of the bicommutants coincide and are the closure of the algebra with respect to some or all of the considered topologies.

Nontrivial zeros of the Racah quadrupole invariant
View Description Hide DescriptionIt is shown that a class of nontrivial zeros of the Racah quadrupole invariant operator is given by two orbits of the group action of an infinite discrete subgroup of the proper two‐dimensional Lorentz group SO(1,1) on the hyperbola 4x ^{2}−3y ^{2}= (11)/(4) .

Symmetry chains and adaptation coefficients
View Description Hide DescriptionGiven a symmetry chain of physical significance it becomes necessary to obtain states which transform properly with respect to the symmetries of the chain. In this article we describe a method which permits us to calculate symmetry‐adapted quantum states with relative ease. The coefficients for the symmetry‐adapted linear combinations are obtained, in numerical form, in terms of the original states of the system and can thus be represented in the form of numerical tables. In addition, one also obtains automatically the matrix elements for the operators of the symmetry groups which are involved, and thus for any physical operator which can be expressed either as an element of the algebra or of the enveloping algebra. The method is well suited for computers once the physically relevant symmetry chain, or chains, have been defined. While the method to be described is generally applicable to any physical system for which semisimple Lie algebras play a role we choose here a familiar example in order to illustrate the method and to illuminate its simplicity. We choose the nuclear shell model for the case of two nucleons with orbital angular momentuml=1. While the states of the entire shell transform like the smallest spin representation of SO(25) we restrict our attention to its subgroup SU(6)×SU(2)_{ T }. We determine the symmetry chains which lead to total angular momentum SU(2)_{ J } and obtain the symmetry‐adapted states for these chains.

Spinor group and its restrictions
View Description Hide DescriptionA realization of the spinor algebra of the rotation group SO(N), N=2n or 2n+1, in the covering algebra of U(2^{ n }) is exploited to obtain explicit representation matrices for the SO(N) generators in the basis adapted to the subgroup chain SO(N)⊃U(n)⊇U(n−1)⊃⋅⋅⋅⊃U(1). As a special case the computation of matrices of U(n) representations characterized by a k‐column Young tableau is reduced to the evaluation of at most k‐box totally symmetric representations of U(2^{ n }).

Grassmann analogs of classical matrix groups
View Description Hide DescriptionIn general, if the parameters of a real or complex algebraic matrix group are replaced by Grassmann parameters, without changing the algebraic constraints, the resulting set fails to form a group. It is shown how to remedy this defect of naive Grassmannification by generalizing the constraint relations. In particular, it is shown how to define Grassmann analogs of the orthogonal, unitary, and symplectic groups.

On the infrared singularity of the resolvent of some Yang–Mills‐type operators
View Description Hide DescriptionFor operators of Yang–Mills type H=∑^{4} _{ j=1}[−i(∂/∂x _{ j })⊗1 +A _{ j }(x)]^{2}+W(x) in L ^{2}(R^{4})⊗V the infrared singularity of the resolvent (H−ζ)^{−} ^{1} is described completely in terms of asymptotic expansions as ζ→0. A _{ j }(x) and W(x) are required to satisfy A _{ j }(x)=O(‖x‖^{−2−δ}), W(x) =O(‖x‖^{−2−δ}), as ‖x‖→∞, δ>0.

Symmetries of the higher‐order KP equations
View Description Hide DescriptionSymmetry generators T ^{(l)} _{ n }=t I _{ n }+L ^{(l)} _{ n } are constructed for the lth‐order Kadomtsev–Petviashvili (KP) equation for all n≥l−2. In the case of the ordinary KP equation (l=2) these symmetries are those found by Chen e t a l. [Physica D 9, 439 (1983)].

Connections on infinitesimal fiber bundles and unified theories
View Description Hide DescriptionWe provide an intrinsic coordinate‐free formalism of Jordan’s version of five‐dimensional Kaluza–Klein or projective theory of relativity in terms of the so‐called infinitesimal fiber bundles, whose structures are slightly more general than principal fiber bundles with connections. Higher‐dimensional generalizations are then suggested, thereby providing a more comprehensive unified theory.

Mapping of connections on bundles and gauge field theories
View Description Hide DescriptionThe problem of solving the combined gravitational and Yang–Mills field systems is regarded as a purely geometrical problem of determining a linear connection on the principal frame bundle L(M) from a connection on a SU(2) principal bundle over a space‐time M. It is suggested that mapping theorems of connections on bundles may provide a means of actually solving ‘‘field equations.’’

A local limit theorem for strongly dependent random variables and its application to a chaotic configuration of atoms
View Description Hide DescriptionWe investigate one‐dimensional chaotic configurations of atoms, which are generated by the Baker transformation or, equivalently, by the Bernoulli shift. The problem of calculating the distribution of the jth nearest‐neighbor distances of these configurations is shown to be equivalent to the task of finding the limit distribution of the sum of the strongly dependent random variables X _{ l }:([0,1),μ_{L})→[0,1), x→(2^{ l } x)mod 1 (l∈N_{0}, μ_{L} is the Lebesgue measure). We prove the validity of a local limit theorem for this sequence of random variables and conclude, therefore, that the distribution density G _{ j } of the jth nearest‐neighbor distances is asymptotically (as j→∞) a Gaussian distribution, the width of which grows as (j)^{1/2}. With the aid of this result, we prove that the pair distribution functionG of our configurations, which is the sum of the G _{ j }’s, tends to unity in the limit of large distances.

Generalization of the diffusion equation by using the maximum entropy principle
View Description Hide DescriptionBy using the so‐called maximum entropy principle in information theory, one derives a generalization of the Fokker–Planck–Kolmogorov equation which applies when the n first transition moments of the process are proportional to Δt, while the other ones can be neglected.

The Gibbs phenomenon in generalized Padé approximation
View Description Hide DescriptionThe Gibbs phenomenon in generalized Padé approximation is discussed, and with the aid of some rational approximants the Gibbs constants are determined. In addition, the steepest of the rational approximants is calculated.

Energy levels of a two‐dimensional anharmonic oscillator: Characteristic function approach
View Description Hide DescriptionThe eigenlevels of a two‐dimensional quartic anharmonic oscillator are identified as the zeros of a ‘‘characteristic function’’ derived from the resolvent. Numerical values are obtained by means of Padé approximants.

Ion‐acoustic dispersion relation with direct fractional approximation for Z′(s)
View Description Hide DescriptionA direct fractional approximation for the derivative Z′ of the plasma dispersion function Z(s) has been obtained by using the modified asymptotic Padé method. The dispersion relation for the ion‐acoustic wave and for the ion‐beam instability has been solved by the use of that fractional approximation yielding satisfactory results. A comparison is given between the dispersion relations calculated with (a) our approximation, (b) other approximations to Z′, and (c) the exact function.

Lie–Bäcklund symmetries of certain nonlinear evolution equations under perturbation around their solutions
View Description Hide DescriptionA generalization of the Lie–Bäcklund (LB) theory for coupled evolution equations is discussed. As a consequence, the direct connection between the LB symmetries and constants of motion of these systems is also established. Furthermore, as an application of this theory we investigate the existence of infinitely many commuting LB symmetries of certain nonlinear evolution equations under perturbation around their solutions. Then the corresponding constants of motion are derived, which are in agreement with the known results.

Theorem on linearized Hamiltonian systems
View Description Hide DescriptionMany nonlinear field equations can be written in Hamiltonian form. Thus the equation ∂_{ t } u=K(u) can be written ∂_{ t } u =[u, H], where H is an appropriate functional and [ , ] is a Poisson bracket. Frequently one is interested in the solution of the equation linearized about a given solution, i.e., the equation ∂_{ t } τ=K′(τ), where K′(τ)=(d/dε) K(u+ετ)‖_{ε=0}. It is known that if a functional I is a constant of motion then τ=[u, I] is a solution. Recently, more general solutions of this form have been found. To prove these results, it is very useful to have the answer to the following question: If I _{ i }, I _{ j } are two functionals, and K _{ i } =[u, I _{ i }], what is K ^{′} _{ i }(K _{ j })? The answer is K _{ i }(K _{ j }) =[[u, I _{ i }], I _{ j } ]. Previously, this was proved assuming that canonical coordinates can be introduced. Here a proof is given without any such assumption.

Exact solution for the spatially inhomogeneous nonlinear Kac model of the Boltzmann equation
View Description Hide DescriptionWe study the spatially inhomogeneous Kac model of the nonlinear Boltzmann equation in 1+1+1 dimensions (velocityv, time t, position x). We obtain an exact solution which is the product of a Maxwellian with a time‐dependent width by a second‐order polynomial in the velocity variable. The solution satisfies a specular reflection condition at the boundary x=x _{0}. The position x _{0}−x appears linearly and only in the odd part of the velocity distribution, the range of x _{0}−x being arbitrarily large but finite in order to maintain the positivity of the distribution. The local density is spatially homogeneous. Further, a particular linear relation between the moments of the cross section must be satisfied. The most general Maxwellian width has two relaxation times and their ratio is a function of the moments of the cross sections. Depending on whether this ratio is larger than or smaller than 1 we find contraction or expansion. The solution relaxes towards a Maxwellian equilibrium solution. Studying the Tjon overpopulation effect of high velocity particles, we find that it depends weakly on the initial condition, and strongly on both the microscopic model of cross section and on the ratio of the two relaxations times. We give a simple criterion (linked to the distinction between contraction and expansion) for the existence of the effect. Theoretically and numerically we test its validity and its failure.

Infinitesimal transformations about soliton solutions of sine–Gordon and modified Korteweg–de Vries equations
View Description Hide DescriptionInfinitesimal transformations about n‐soliton solutions of sine‐Gordon and modified Korteweg–de Vries equations are obtained using respective Bäcklund transformations. We also obtain the eigenfunctions of corresponding generalized Zakharov–Shabat systems.