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Volume 21, Issue 4, April 1980

Generalized Grassmann algebras with applications to Fermi systems
View Description Hide DescriptionGeneralized Grassmann numbers x _{ i } (i=1,2,⋅⋅⋅,n) are defined as those satisfying the relations x _{ i } x _{ j }=η_{ i j } x _{ j } x _{ i } with η_{ i j } =−(+ or −) for i=j(i≠j). Ordinary Grassmann numbers correspond to a special case η_{ i j }=− for all i, j. Mathematical properties of such numbers are discussed in detail, and it is found that most of the results known for the ordinary case can naturally be extended to the general case. Applications are made to a description of general Fermi systems where the commutation relations belong to an arbitrary anomalous case.

Para‐Grassmann algebras with applications to para‐Fermi systems
View Description Hide DescriptionGrassmann algebras of the usual kind are generalized to what are to be called para‐Grassmann algebras. Para‐Grassmann numbers are defined as those satisfying the trilinear commutation relations that resemble the para‐Fermi commutation relations. Basic mathematical properties of these algebras are studied in detail on the basis of generalized Grassmann algebras discussed in the preceding paper. Applications are made to description of para‐Fermi systems, and it is found that such systems can completely be described in what we call the para‐Grassmann representation.

A Galileian formulation of spin. II. Explicit realizations
View Description Hide DescriptionThis paper is the follow‐up of an earlier one. It reviews some of the lesser known ideas involved in the representation theory of Clifford algebras and applies these ideas in computing explicit realizations of the spin groups and the Lie algebras appearing in the previous paper.

A note on the symmetries of the 3j and 6j coefficients. I
View Description Hide DescriptionIt is shown that the study of the symmetries of the 3j coefficient in terms of the set of six _{3} F _{2}(1)’s derived by us introduces a six‐to‐one homomorphism of the 72‐element group of symmetries of the 3j coefficient on to the 12 permutations of parameters of a single _{3} F _{2}(1) series of the set. Also, the study of the symmetries of the 6j coefficient in terms of the set of three (four) _{4} F _{3}(1)’s derived by us introduces a three (four) ‐to‐one homomorphism of the 144‐element group of symmetries of the 6j coefficient on to the 48(36) allowed permutations of parameters of a single _{4} F _{3}(1) series of the set.

The most degenerate irreducible representations of the symplectic group
View Description Hide DescriptionMatrix elements of the generators of the symplectic group Sp(2n) have been obtained for the most degenerate irreducible representations (m _{2n },?) and (?_{2n }). These are the only two cases for n≳2 where the state labels according to the branching laws of Hegerfeldt give rise to an orthogonal set of basic vectors.

The Lorentz group in the oscillator realization. II. Integral transforms and matrix elements of SO(2,1)
View Description Hide DescriptionWe use the boson realization to investigate the connection between integral transforms and matrix elements of SO(2,1) in a continuous basis for both the principal and discrete series of representations. Matrices in the SO(1,1) basis are shown to be related to an integral transform of Mellin–Barnes type.

Permutational symmetry of many particle states
View Description Hide DescriptionThe symmetry of the Nth rank tensor basis for an irreducible representation of U (n) under the operations of the permutation group has been investigated. It has been found that symmetrized linear combinations of the elements of the matrix algebra of S _{ N } lead to a tensor basis for U (n) yielding the same matrix elements as the Gel’fand–Tsetlin basis for the generators E _{ i,i±1} of U (n). Based on these developments an algorithm has been developed for directly determining the matrix elements of the generators E _{ i j }(j≠i±1) of U (n) using a pattern calculus.

Particle statistics from induced representations of a local current group
View Description Hide DescriptionRepresentations of the nonrelativistic current group S‐K are studied in the Gel’fand–Vilenkin formalism, where S is Schwartz’ space of rapidly decreasing functions, and K is a group of diffeomorphisms of R^{ s }. For the case of N identical particles, information about particle statistics is contained in a representation of K_{ F } (the stability group of a point F∈S′) which factors through the permutation group S _{ N }. Starting from a quasi‐invariant measure μ concentrated on a K orbit Δ in S′, together with a suitable representation of K_{ F } for F∈Δ, sufficient conditions are developed for inducing a representation of S‐K. The Hilbert space for the induced representation consists of square‐integrable functions on a covering space of Δ, which transform in accordance with a representation of K_{ F }. The Bose and Fermi N‐particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of S‐K determines a well‐defined representation of K_{ F }; (2) equivalent representations of S‐K determine equivalent representations of K_{ F }; (3) a representation of K_{ F } induces a representation of S‐K; and (4) equivalent representations of K_{ F } determine equivalent induced representations.

On the dynamical symmetries of the Kepler problem
View Description Hide DescriptionWe try to understand the geometry of the SO(n+1,2) action on the Kepler Manifold of the n‐dimensional hydrogen atom. We show that the SO(n+1,2) symmetry of the Kepler Problem is closely related to the fact that the geodesic flow on T*S ^{ n } is periodic. We also exhibit the orbit picture analog of the peculiar property of the corresponding SO(n+1,2) representation; that is, it stays irreducible when restricted to SO(n+1,1) subgroups.

Missing label operators in the reduction Sp(2n) ↓Sp(2n−2)
View Description Hide DescriptionI consider the ’’missing label’’ problem for basis vectors of an Sp(2n) representation corresponding to a group reduction chain with links Sp(2ν) ↓Sp(2ν−2) ×U(1), 2⩽ν⩽n. I obtain two different sets of ν−1 missing label operators. These operators, together with the ν Casimir operators of the Sp(2ν) group, the ν−1 Casimir operators of the Sp(2ν−2) subgroup and the one generator of the U(1) subgroup, form a complete set of labeling operators whose eigenfunctions provide a canonical basis in the representation space of Sp(2ν). When the number of missing labels exceeds one the most general solution to the labeling problem is not known. The two particular solutions presented here have certain appealing aspects of symmetry and simplicity.

On the invariant scalar products and the UIR of SO (n,1)
View Description Hide DescriptionA quantity, which is invariant under the transformation of h∈SO (n,1), is found in a form of the series expansion in terms of the D matrix elements of SO (n) and an invariant scalar product, which is needed for the complementary series of the unitary irreducible representation of SO (n,1), is introduced in the Hermitian form by using the invariant quantity. The action of the intertwining operator defined through the Hermitian scalar product to the bases of H _{ c }, on which the complementary series of the UIR of SO (n,1) are realized, is found.

Canonical transforms. IV. Hyperbolic transforms: Continuous series of SL(2,R) representations
View Description Hide DescriptionWe consider the sl(2,R) Lie algebra of second‐order differential operators given by the Schrödinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the C ^{ε} _{ q } continuous series of representations. The corresponding SL(2,R) Lie group is shown to be a group of integral transforms acting on a (two‐component) space of square‐integrable functions, with an integral (matrix) kernel involving Hankel and Macdonald functions. The subgroup bases for irreducible representations consist of Whittaker, power, Hankel, and Macdonald functions. We construct the operator which intertwines this realization of SL(2,R) with the more familiar Bargmann realization on functions on the unit circle. This operator implements the canonical transformation of the above Schrödinger systems to action and angle variables.

Classification of real simple Lie superalgebras of classical type
View Description Hide DescriptionFinite‐dimensional simple Lie superalgebras (also called Z_{2}‐graded Lie algebras) over an algebraically closed field of characteristic zero were classified in 1976. All simple Lie superalgebras over the reals, whose Lie subalgebra is reductive, are determined here up to isomorphism. As is the theory of simple Lie algebras, this is done by classifying the involutive semimorphisms of the complex Lie superalgebras. One sees in particular that the real form of the Lie subalgebra completely determines the real form of the Lie superalgebra.

Ewald evaluation and Poisson summation formulas for a class of near‐Coulombic lattice sums
View Description Hide DescriptionThe class of lattice sums studied is defined by F _{ n,τ}=∑^{∞} _{τ}′[(1/τ^{ n }) −(1/Ω) ∫_{τ}(d^{3} r/r ^{ n })] −(1/Ω) ∫_{0}(d ^{3} r/r ^{ n }) for 1⩽n⩽2, where {τ⃗} denotes a three‐dimensional Bravais lattice with volume Ω per lattice point, {γ⃗} denotes its reciprocal lattice normalized with exp(iτ⃗⋅γ⃗) =1 with volume v= (2π)^{3}/Ω per lattice point, and except for the case n=1 the cellular integrations are taken over centered cells, proximity or primitive. Ewald evaluation is discussed for all cases and reciprocal relations between the F _{ n,τ} and the F _{(3−n),γ} are supplied for all n. These relations are essentially Poisson summation formulas (PSF) with modifications for the cases n=1,2. The method of securing the PSF’s involves a direct, and an inverse, use of an Ewald method coupled with careful analysis of all interchanges of integrations and summations. The method should be useful for similar studies on other complex sums which, like these, are not very amenable to direct application of a PSF with assurance of validity.

Sum rules for zeros of polynomials. I
View Description Hide DescriptionIt is shown that for polynomials satisfying differential equations of a particular form it is easy to generate sum rules for the powers of the zeros. All of the classical orthogonal polynomials are of this form. Examples are given for the Hermite, Laguerre, Tchebycheff, and Jacobi polynomials. In particular an explicit formula is given for the sums of all powers of Tchebycheff zeros. This same formula gives the sums for general Jacobi polynomials in the limit of large N.

Sum rules for zeros of polynomials. II
View Description Hide DescriptionSum rules for zeros of polynomials which satisfy differential equations with polynomial coefficients of order no higher than the derivative they multiply are extended. First, higher order coefficients are considered. Second, sum rules for negative powers are obtained. Finally, the results are extended to a class of integral functions.

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. I
View Description Hide DescriptionWe develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scatteringtransforms and nonlinear ordinary differential equations (ODE) of P‐type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P‐type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.

On Mathieu equation with damping
View Description Hide DescriptionA direct variational method is applied to the linear and nonlinear Mathieu equation with damping. It is found that the nature of the periodic solutions and the characteristic curves are modified due to the presence of the damping. A threshold value of β is required to overcome the damping for the existence of the periodic solutions. Stability analyses for the periodic solutions are also carried out.

On the separability of the sine‐Gordon equation and similar quasilinear partial differential equations. II. Dependent‐ and independent‐variable transformations
View Description Hide DescriptionIn an earlier paper, we investigated the separability of the sine‐Gordon equation (SGE), and of similar quasilinear partial differential equations, under transformations of the dependent variable (i.e, of the codomain). We found, in particular, that there is a general class of dependent‐variable transformations which leads to separable forms of the SGE. In this paper, we extend our previous analysis to include independent‐ as well as dependent‐variable transformations (i.e., transformations of both the domain and codomain) and treat, in detail, constant coefficient equations of the first and second orders. We illustrate our method by applying it to the SGE and find combinations of domain and codomain transformations which reduce the equation to separable forms. Some of these transformations lead to known solutions of the SGE, but others give new solutions expressible in terms of a fifth Painlevé transcendent. Our method can, in principle, be used to map out the space of separable solutions of the SGE and other similar second‐order equations, but it does have limitations. A discussion of these limitations and suggestions for possible improvements are also given.

Construction of the phase of the scattering amplitude by iteration of bounds from unitarity integral
View Description Hide DescriptionGiven the modulus of the amplitude a convergent iteration scheme is given to obtain the phase from the unitarity equation. The method is based on improving the upper and lower bounds on the phase. Uniqueness of the solution is investigated and numerical results for a specific case are given.