Volume 24, Issue 4, April 1983
Index of content:

Weight vector representatives for SO(2N)
View Description Hide DescriptionFollowing Mohapatra and Sakita, it is convenient to rewrite the algebra of 2Nγ‐matrices in terms of N annihilation operators and N creation operators. We find them useful in representing weight vectors in the spinorial σ_{ N } σ_{ N } ^{c} and the irreducible subcomponents of σ_{ N }×σ_{ N } and σ_{ N }×σ_{ N } ^{c} operators. Though their Dynkin labels are readily accessible, a natural basis for the enumeration of their complete set of weights is given by N positive weights of the vectorial representation. Their subgroup content under SU(N)⊆SO(2N) and SO(2m)⊗SO(2N−2m)⊆SO(2N) is made obvious by using a simple identity. Conjugation and Yukawa couplings are touched on briefly.

Cartesian polytensors
View Description Hide DescriptionA Cartesian polytensor is defined as a set of Cartesian tensors in a sequence of increasing rank. A matrix formulation of polytensors is given to express arrays of direct tensor products and series of tensor contractions in concise form. The transformation of a polytensor under rotation of coordinate axes is shown to be accomplished by means of an orthogonal matrix. The special properties of compressed polytensors, composed of totally symmetric tensors with redundant components deleted, are demonstrated. The use of polytensors is illustrated by an application to the problem of interactions among polarizable electric charge distributions.

Some character theory for groups of linear and antilinear operators
View Description Hide DescriptionElementary group concepts are recast into a form applicable to finite magnetic groups of linear and antilinear operators. Analogs of useful definitions for linear groups such as the Frobenius–Schur invariant, commutator subgroups, and ambivalent classes are considered. These are applied to the 180 magnetic single and double point groups and it is shown that only seven require independent treatment of characters.

On the 3j symmetries
View Description Hide DescriptionThe symmetry properties of the 3j mtensor for any finite or compact linear group are discussed using a wreath product construction. This is shown to provide a complete group theoretic explanation for all symmetry properties whether ‘‘essential’’ or ‘‘arbitrary.’’ The link with the similar—but distinct—method of inner plethysms is considered.

A theorem on orbit structures (strata) of compact linear Lie groups
View Description Hide DescriptionWe present a comprehensive constructive proof of a theorem characterizing the tangent space to a stratum (orbit structure) of the Euclidean spaceR ^{ n }, seat of an orthogonal representation of a compact group G. The characterization is made in terms of gradients of a complete set (integrity basis) of G‐invariant polynomials. In a recent paper [M. Abud and G. Sartori, Phys. Lett. B 1 0 4, 147 (1981)], the theorem, which may be considered a generalization of a theorem by Michel [C. R. Acad. Sci. Ser. A 2 7 2, 433 (1971)], has been shown to be effective in the determination of the equations of the strata and in the determination of natural extrema of G‐invariant functions.

Internal labels of degenerate representations
View Description Hide DescriptionAn expression for the number of internal labels of degenerate irreducible representations of compact semisimple Lie groups is given in the same spirit as Racah’s formula for the nondegenerate case.

Analysis of the outer product for the symmetric group
View Description Hide DescriptionExpressions are derived to write the basis vectors for an irreducible representation μ of the symmetric group in terms of basis vectors for irreducible representations whose outer product yields μ.

Complementary group with respect to SO(n)
View Description Hide DescriptionWe look for a complementary group with respect to SO(n) within either irreducible representation 〈(1/2)^{ d n }〉 or 〈(1/2)^{ d n−1}(3/2)〉 of the group Sp(2d n,R) of linear canonical transformations in a 2d n‐dimensional phase space. We prove that: (i) such a group is Sp(2d,R) when n=2q+1 or n=2q>2d; (ii) it is SU(d,d) when n=2 and d>1; (iii) it does not exist when n=2, d=1, or 2<n=2q≤2d.

Some special SU(3)⊇R(3) Wigner coefficients and their application
View Description Hide DescriptionBargmann space expansions of oscillator functions are used to derive analytic expressions for SU(3)⊇R(3) Wigner coefficients for the couplings (λ_{1}0)×(0μ_{2})→(λ_{3} μ_{3})L _{3}=0 and (λ_{1}0)×(λ_{2}0)→(λ_{3} μ_{3})L _{3}=0, with arbitrary (λ_{3} μ_{3}). These lead to expansions useful in nuclear cluster problems and are used to give a simple form for the SU(3)‐irreducible tensor expansion of a scalar two‐body interaction, an application which motivated this investigation.

Reciprocal transformation for one‐dimensional conservation equations
View Description Hide DescriptionOne‐dimensional conservation equations (OCE) of the form ∂n/∂t+∂f/∂x=0 with n=n(x,t)>0 and f=f(n,∂n/∂x, ∂^{2} n/∂x ^{2},⋅⋅⋅) admit a symmetric r e c i p r o c a l t r a n s f o r m a t i o n x→x*(x,t), n→n*(x*,t)≡n ^{−} ^{1}, f→f*≡−n ^{−} ^{1} f, which produces an equivalent OCE for n* in x* space. Certain OCE of contemporary interest are r e c i p r o c a l i n v a r i a n t in the sense that f*=f(n*, ∂n*/∂x*, ∂^{2} n*/∂x*^{2},⋅⋅⋅). There also exists a class of essentially nonlinear OCE for which the reciprocal transformation produces a linear OCE, and thus equations in this class are solvable analytically.

Lie transformations, nonlinear evolution equations, and Painlevé forms
View Description Hide DescriptionWe present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one‐parameter continuous (Lie) group of transformations. It is shown that, in general, the corresponding invariant variables (the subclass of which is the usual similarity variables) lead to ordinary differential equations of Painlevé type in the case of inverse scattering transform solvable equations, as conjectured by Ablowitz, Ramani, and Segur. This is found to be also true for certain higher spatial dimensional versions such as the Kadomtsev–Petviashivilli, two dimensional sine–Gordon, and Ernst equations. For the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points.

Kaluza–Klein theories on bundles with homogeneous fibers. I
View Description Hide DescriptionWe analyze some geometric aspects of Kaluza–Klein theories under the assumption that the (4+d)‐dimensional space is a bundle over space–time M with fiber G/H. We formulate the most general metric in the bundle which leads, upon dimensional reduction of the Ricci scalar, to a G‐gauge invariant Lagrangian. We find that the treatment of Brans–Dicke‐like scalars given by some authors is inconsistent with the bundle‐theoretic interpretation.

Error estimates of solutions and mean of solutions of stochastic differential systems
View Description Hide DescriptionStochastic differential equations are considered. Estimates in terms of statistical properties are given for the difference between the solutions and solutions of the mean of stochastic differential systems. For this purpose necessary theorems are developed and sufficient conditions are given to obtain error estimates. A few examples are worked out to demonstrate the usefulness of the results.

Explicit suspensions of diffeomorphisms—An inverse problem in classical dynamics
View Description Hide DescriptionPresented in this paper is a set of explicit prescriptions for associating with a given map of R ^{ n }, which is C ^{2}‐isotopic to the identity, a time‐dependent vector field whose time‐1 map is the given one. Also shown is how to apply additional restrictions to the vector field including that it be (1) periodic in time, (2) Hamiltonian, and (3) of potential form; several examples show numerical verification of the theory.

Inverse problem for the reduced wave equation with fixed incident field. III
View Description Hide DescriptionThe inverse problem for the reduced wave equation Δu+k ^{2} n ^{2}(x)u=0, x∈R ^{3}, is examined for the case where measurements of the amplitude of the scattered field (produced by a fixed incident field at a single frequency) are obtained at a finite number of points. A strategy is given for the recovering of the phase data through the minimization of a quadratic form involving comparison data. The problem is then reduced to the problem treated in previous papers where the complex‐valued quantities u ^{s}(x _{ l }) are known at a finite number of points. A relationship between the smallest eigenvalue of the ‘‘measurement’’ matrix and ∥K∥_{2} is given.

Operator formalism equivalent to the Feynman quantization technique
View Description Hide DescriptionThe concept of a Fock–Stueckelberg space of quantum states and a procedure of an operator quantization using only Lagrangians (kinematical quantization) are introduced. A propagator operator K, matrix elements of which are Green’s functions, is used, and an equation of motion for it is derived. We prove that kinematical quantization is an operator (coordinate‐free) form of the Feynman quantization technique. The Feynman path integral (FPI) is obtained as a spectral representation of the operator K in a coordinate basis. The connection of a representation of commutation relations in this scheme, the domain of integration in FPI, and causality is mentioned.

Weak dispersion‐free states and the hidden variables hypothesis
View Description Hide DescriptionWe investigate dispersion‐free states which are additive only on the pairs containing a central element (central‐absolutely compatible). We show that any logic possesses plenty of such states, in fact, as many as a certain Boolean algebra. The latter result matches the hidden variables conjecture.

A generalization of Cohen’s theorem
View Description Hide DescriptionThe theorem due to L. Cohen, which implies that quantum mechanics cannot be formulated as a stochastic theory in phase space, is generalized. The assumption that the phase‐space representatives of the density operators satisfy the quantum mechanical marginals is replaced by the weaker condition of Ω‐representability.

Path integration of an action related to an electron gas in a random potential
View Description Hide DescriptionPath integration of an action related to an electron gas in a random potential is performed within the framework of Feynman’s polygonal path approach. The exact propagator obtained is simply related to the harmonic oscillator propagator. The integration is direct and does not require the knowledge of an auxiliary measure or the artificial coupling of the system to the external forces.

A partial inner product space of analytic functions for resonances
View Description Hide DescriptionGeneralized Hilbert spacesD(α,β) are defined using analytic continuation of Hardy class functions into a wedge bounded by the angles α,β. Eigenfunctions of i s o l a t e d complex eigenvalues may be found in D(α,β) for operators that have a self‐adjoint representation in L ^{2}. These eigenvalues correspond to resonances in the associated decay problem. A bilinear form between D(α,β) and D(−β,−α) is defined, which has some of the properties of a Hilbert space scalar product, and it is shown that this form can be used to define a variational principle to obtain the eigenvalue equations.