Volume 18, Issue 6, June 1977
Index of content:

Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(n)
View Description Hide DescriptionWe study the space P of all polynomialfunctions on the complex cone K_{ n }={Z= (Z_{1}⋅⋅⋅Z_{ n }) ε C^{ n }, Z^{2}=Z^{2} _{1}+ ⋅⋅⋅+Z^{2} _{ n }=0} (n=3,4,⋅⋅⋅). Its subspacesK ^{ l } (=K ^{ l } _{ n }) of homogeneous polynomials of degree l (=0,1,2,⋅⋅⋅) provide a convenient realization of the carrier spaces for the symmetric tensor representations of the real orthogonal group SO(n). The multiplication operator Z_{μ} (μ=1,...,n) maps K ^{ l } into K ^{ l+1}. We define its adjoint as an interior differential operator on K_{ n } which maps K ^{ l+1} into K ^{ l } and transforms as an n‐vector. We show that the lowest order differential operator with this property is proportional to D _{μ}= (n/2−1+√∂) ∂_{μ} −(1/2) Z_{μ}Δ. We define a scalar product in P with respect to which the operators Z_{μ} and D _{μ} are Hermitian adjoint to each other and consider the Hilbert space completion K _{ n } of P with respect to this scalar product. The spaces K _{ n } are imbedded for all n (=3,4,⋅⋅⋅) in the Fock type spaces B _{ n }, studied earlier by Bargmann. The space K _{ n } possesses a reproducing kernel that allows us to define a (unique) harmonic extension of every analytic function in K _{ n }. It is shown that the spaces K _{3} and K _{4} can be imbedded isometrically in the Hilbert spacesB _{2} and B _{4} associated with the representations of SU(2) and SU(2) ×SU(2) [⊇SU(4)].

Conformal properties of a class of exactly solvable N‐body problems in space dimension one
View Description Hide DescriptionAn algebra of collective variables in the generalized (Calogero–Sutherland) N‐body classical one‐dimensional model is introduced, and their transformation properties under the conformal group are discussed in detail.

Conditional expectations in generalized probability theory
View Description Hide DescriptionExpectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub‐σ‐algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.

Highest weights of semisimple Lie algebras
View Description Hide DescriptionThe nine well‐known semisimple Lie algebras are partitioned in two classes: W _{ l p c=1} (all roots have the same length) and W _{ l z c≠1} (the roots have two different lengths of ratio equal to c ^{1/2}). For each of these two classes a general expression is given for few elements of interest as the highest weight vector (h.w.v.) L and its power δ (L), the eigenvalues of the second order Casimir operator, the width of a weight diagram, the dimensions and the matrix elements of irreducible representations of the algebras. In the Appendix are given two examples of application of this paper.

On a topological problem arising in physics
View Description Hide DescriptionThe investigation of a large class of problems in physics requires the determination of the ranges of some parameters ξ, E, ⋅⋅⋅ for which equations of the form F (r; ξ, E,⋅⋅⋅) =0 have some or no roots in a given bounded or unbounded interval of r. The solution of this problem is given under the form of three general theorems. Examples of utilization in physics are also discussed.

Kerr’s theorem and the Kerr–Schild congruences
View Description Hide DescriptionA simple flat spacetime derivation of the Kerr–Schild congruences is presented starting from Kerr’s theorem.

Composite next nearest neighbor degeneracy
View Description Hide DescriptionExpressions are derived which yield, exactly, the composite degeneracy of those arrangements of simple, indistinguishable particles on a one‐dimensional lattice space, which are characterized by the number of nearest and next nearest neighbor pairs.

Exact nearest neighbor statistics for λ‐bell particles on a one‐dimensional lattice
View Description Hide DescriptionRelationships are developed which describe exactly the degeneracies of those states specified by the number of either the occupied, mixed, or vacant nearest neighbor pairs which exist when indistinguishable λ‐bell particles (occupying λ contiguous sites) are distributed on a one‐dimensional lattice space.

Killing inequalities for relativistically rotating fluids. II
View Description Hide DescriptionTheorems that the angular momentum density and the angular velocity of locally nonrotating observers are postive are established for general relativistic fluids which rotate differentially with positive angular velocity. The results apply to the pseudostationary case in which the system possesses an ergoregion. An application to the stability of ergoregions is discussed.

Inequivalent sets of commuting missing label operators for SU(4) ⊆SU(2) ×SU(2)
View Description Hide DescriptionWe exhibit one possible choice for the four functionally independent label operators available. We prove that they can be separated into two inequivalent, i.e., functionally independent, sets of commuting label operators, namely the set of operators Ω and Φ first considered by Moshinsky and Nagel, and the set of operators C ^{(202)} and C ^{(022)}.

Evolution equations possessing infinitely many symmetries
View Description Hide DescriptionA general method for finding evolution equations having infinitely many symmetries or flows which preserve them is described. This is applied to the Korteweg–de Vries, modified Korteweg–de Vries, Burgers’, and sine–Gordon equations.

Symmetries of ultralocal quantum field theories
View Description Hide DescriptionThe symmetries of the gradient free ultralocal modelquantum field theories are studied. The internal parameter λ introduced by J. R. Klauder is replaced by an r‐component vector and used to obtain an r‐vector ultralocal field operator φ̃. Then ultralocal many‐body models with translational and rotational symmetry are set up by partitioning λ⃗ into N, 3‐d ’’relative internal coordinates.’’ Hartree‐type 1/N limits are studied in this model and found to be accurate only if contributions from ‖χ⃗‖≳≳0 are negligible. A brief sketch is given of how to produce more general U _{ N } and pairing interaction ultralocal models.

Axiomatic basis for spaces with noninteger dimension
View Description Hide DescriptionFive structural axioms are proposed which generate a space S_{ D } with ’’dimension’’ D that is not restricted to the positive integers. Four of the axioms are topological; the fifth specifies an integration measure. When D is a positive integer, S_{ D } behaves like a conventional Euclidean vector space, but nonvector character otherwise occurs. These S_{ D } conform to informal usage of continuously variable D in several recent physical contexts, but surprisingly the number of mutually perpendicular lines in S_{ D } can exceed D. Integration rules for some classes of functions on S_{ D } are derived, and a generalized Laplacian operator is introduced. Rudiments are outlined for extension of Schrödinger wave mechanics and classical statistical mechanics to noninteger D. Finally, experimental measurement of D for the real world is discussed.

A stochastic derivation of the Klein–Gordon equation
View Description Hide DescriptionSeveral years ago Nelson succeeded in deriving the nonrelativistic Schrödinger equation within a stochastic model which included Newton’s second law as the fundamental dynamical rule. Unfortunately, the relativistic extension of Nelson’s work is not so straightforward as might at first be supposed. This paper examines the difficulties inherent in such a relativization and proposes supplemental axioms which resolve those difficulties. A stochastic derivation of the Klein–Gordon equation is then presented.

Variational methods for chemical and nuclear reactions
View Description Hide DescriptionAll the variational functionals are derived which satisfy certain criteria of suitability for molecular and nuclear scattering, below the threshold energy for three‐body breakup. The existence and uniqueness of solutions are proven. The most general suitable functional is specialized, by particular values of its parameters, to Kohn’s tanη, Kato’s cot(η−ϑ), the inverse Kohn coη, Kohn’s S matrix, our S matrix, Lane and Robson’s functional, and several new functionals, an infinite number of which are contained in the general expression. Four general ways of deriving algebraic methods from a given functional are discussed, and illustrated with specific algebraic results. These include equations of Lane and Robson and of Kohn, the fundamental R matrix relation, and new equations. The relative configuration space is divided as in the Wigner Rmatrix theory, and trial wavefunctions are needed for only the region where all the particles are interacting. In addition, a version of the general functional is presented which does not require any division of space.

Invariants for the time‐dependent harmonic oscillator
View Description Hide DescriptionLewis showed that in the case p (t) ≡1, h=1, I (t) = (1/2) {p ^{2}(t)[ρ (t) y′ (t) −y (t) ρ′ (t)]^{2}+h ^{2} y ^{2}(t)/ρ^{2}(t) } is constant in time if y (t) solves (p (t) y′) ′+q (t) y =0, and ρ (t) solves p (t) ρ^{3}(t) L[ρ]=h ^{2} (h constant). Recently, Eliezer and Gray showed that I (t) =const is just the conservation of angular momentum in an appropriate physical interpretation. We show, using a change of variable technique, that I (t) =const reduces to sin^{2}ϑ+cos^{2}ϑ=1. We discuss uniqueness and extendability of solutions to the above equation in ρ.

Localized solutions of a nonlinear electromagnetic field
View Description Hide DescriptionThe electronlike localized solutions, for a nonlinear electromagnetic field obtained from the Born–Infeld Lagrangian, are studied.

Pseudoparticle configurations in two‐dimensional ferromagnets
View Description Hide DescriptionIt is proved that all the finite energy solutions to the field equations of the two‐dimensional Heisenberg ferromagnet theory are topologically stable.

Time‐dependent dynamical symmetries and constants of motion. III. Time‐dependent harmonic oscillator
View Description Hide DescriptionThis paper is a continuation of previous Papers I and II [J. Math. Phys. 17, 1345 (1976); 18, 424 (1977)]. In the present paper we apply the theory (based upon Lagrangian dynamics) developed in I and II to obtain the dynamical symmetries and concomitant constants of motion admitted by the time‐dependent n‐dimensional oscillator (a) E ^{ i }≡χ̈^{ i } +2ω (t) x ^{ i }=0. The dynamical symmetries are based upon infinitesimal transformations of the form (b) χ̄^{ i }=x ^{ i }+δx ^{ i }, δx ^{ i }≡ξ^{ i }(x,t) δa; =t+δt, δt≡ξ^{0}(x,t) δa which satisfy the condition (c) δE ^{ i }=0, whenever E ^{ i }=0. It is shown that such symmetries of the oscillator (a) will be time‐dependent projective collineations. For such symmetries which satisfy the R _{1} restriction (defined in I) it is shown there exist concomitant constants of motion C _{1} of the oscillator, which for n=1 are time‐dependent cubic polynomials in the χ̇ variable, and for n⩾2 are time‐dependent quadratic polynomials in the χ̇^{ i } variables. It is shown that those symmetries which satisfy the R _{2} restriction (Noether symmetry condition discussed in I) are time‐dependent homothetic mappings consisting of time‐dependent scale changes, time‐dependent translations, and rotations. The concomitant Noether constants of motion C _{2} are time‐dependent quadratic polynomials in the χ̇^{ i } variables for all n. The Noether constant of motion C _{2} [referred to as C _{2}(B)] for which the associated underlying symmetry mapping is the time‐dependent scale change is shown to include as a special case when n=1 a class of invariants formulated by Lewis [Phys. Rev. Lett. 18, 510 1967)] (by means of a phase space analysis which applies Kruskal’s theory in closed form). For the case of general n it is shown that the time‐dependent symmetric tensor constant of motion I _{ i j } constructed by Günther and Leach [J. Math. Phys. 18, 572 (1977)] is included as a special case of a time‐dependent symmetric tensor constant of motion K _{ i j }, where K _{ i j } is obtained by use of a time‐dependent related integral theorem by means of the symmetry deformation of the constant of motion C _{2}(B) with respect to the affine collineations; such collineations are a subset of the projective collineation symmetries mentioned above. The symmetries and their concomitant constants of motion of the oscillator (a) with ω (t) of the form ω (t) =a+b e ^{ c t } are obtained.

Eigenfrequency density oscillations and Walfisz lattice sums
View Description Hide DescriptionWe show that recent results on the distributions of eigenfrequencies for the scalar and electromagnetic wave equations in a cube‐shaped domain were anticipated by the work of Walfisz on the number of lattice points in a sphere.