Index of content:
Volume 25, Issue 5, May 1984

On some partitions of hypergraphs and cumulants having applications in statistical mechanics
View Description Hide DescriptionA h y p e r g r a p h H={E _{1},E _{2},...,E _{ n }} is a set of sets; the sets E _{ i } are called the e d g e s, and the elements of V(H)≡U^{ n } _{ i=1} E _{ i } the v e r t i c e s o f H(if each edge contains only two vertices, we have an ordinary graph). Hypergraphs and c u m u l a n t s over hypergraphs occur in various statistical mechanical problems. A cumulant over H involves a sum over partitions of the set H, whence the motivation for studying partitions of hypergraphs and related notions. Writing E _{ i } ‐ ‐ ‐ ‐^{ k‐(v)} E _{ j } if there exist at least k∈N vertex‐disjoint paths between the edges E _{ i } and E _{ j }, we show that ‐ ‐ ‐ ‐^{ k‐(v)} is an equivalence relation on H, and denote by P̂_{ k } H the partition of H into its (‐ ‐ ‐ ‐^{ k‐(v)}) equivalence classes. H is said to be k‐(v) c o n n e c t e d if every pair of edges is in the relation ‐ ‐ ‐ ‐^{ k‐(v)}; we show that there exists a c o a r s e s t partition of H into k‐(v) connected blocks, and denote it by P̂^{′} _{ k } H. Given a partition P(H)={H _{1},H _{2},...,H _{ p }} of H, we denote, for any real number κ, σ_{κ}[P(H)]≡∑^{ p } _{ j=1} (‖V(H _{ j })‖ −κ), σ_{κ}[P(H)] ≡∑^{ p } _{ j=1}(‖V(H _{ j })‖ −κmkP̂_{1} H _{ j }‖), where ‖V(H _{ j })‖ is the number of vertices and ‖P̂_{1} H _{ j }‖ the number of connected components of H _{ j }. We introduce ‘‘subcumulants’’ over H, which involve only partitions of H for which σ^{′} _{κ} has the same value. The subcumulants corresponding to m i n i m u m values of σ_{κ}, κ∈R, are of practical interest (especially the cases κ=0 and 1). We accordingly study the set π^{′0} _{κ}(H) of partitions which minimize σ_{κ}; this is simply related to the set π^{0} _{κ}(H) of partitions which minimize σ_{κ}. We show that P(H)≤P̂^{′} _{ k } H ≤P̂_{ k } H for all P(H)∈π^{0} _{ k }(H) (where ≤ signifies ‘‘is a subpartition of’’), that P≤P′ for every P∈π^{0} _{ k }(H) and P′∈π^{0} _{κ−ε}(H), ε>0, and that π^{0} _{κ}(H) is a sublattice of the (≤) lattice of all partitions of H. The sets π^{0} _{κ}(H) and π^{0} _{κ}(H), and the corresponding minimal values of σ_{κ} and σ^{′} _{κ}, are explicitly determined, for any hypergraph if κ≤1, and for some special types of hypergraphs if κ>1. We introduce a new notion of connectedness for hypergraphs, the σ c o n n e c t i v i t y Σ(H)≡Max{κ,{H}∈π^{0} _{κ}(H)}, and relate it to the v e r t e x c o n n e c t i v i t y k _{ v }(H)≡Max{k, H is k‐(v)connected}; we have, in particular, Σ(H)≤k _{ v }(H) (Σ=k _{ v } if k _{ v }=0 or 1).

Determination of point group harmonics for arbitrary j by a projection method. I. Cubic group, quantization along an axis of order 4
View Description Hide DescriptionWe give a method for systematically building cubic harmonics, i.e., base vectors of irreducible representations of the group of invariance of a cube. Applications include solid state physics, molecular chemistry, spectroscopy, etc. The only necessary operators are C ^{ z } _{4} and R _{ y }, rotation of π/2 around 0y. Projectors on representations are constructed; then their rows are orthogonalized by a Schmidt method and normalized simply by dividing by the square root of a diagonal element.

Determination of point group harmonics for arbitrary j by a projection method. II. Icosahedral group, quantization along an axis of order 5
View Description Hide DescriptionThe method described in the previous paper to obtain cubic harmonics is applied to the similar problem for the icosahedral group. The projection operators are expressed in terms of rotation matrix elements of R(π,φ,0) and R(0,π−φ,π) φ=arctan 2, acting on subspaces invariant under D _{5}.

On a class of spinor representations of SO(7)
View Description Hide DescriptionThe reduction of the irreducible representations [v,0,...,0,1] of SO(2n+1) with respect to SU(2)⊗SU(2)⊗SO(2n−3) is considered. For the n=3 case all the reduced matrix elements of the SO(7) generators in the [SU(2)]^{2}⊗SO(3) basis are calculated with the use of recursion relations.

Multispinor basis for representations of SO(N)
View Description Hide DescriptionThe generators of the rotation groups SO(N) (N=2n, 2n+1) have been realized using a restriction of the unitary group U(2^{ n }) defined on the 2^{ n } ‐dimensional fundamental representation space of spinors. These generators have been used to subduce multispinor representations of SO(N) from those of U(2^{ n }). The procedure has been illustrated for the two‐spinor vector representations 〈10〉 and 〈1000〉 of SO(5) and SO(8), respectively.

Group representations in indefinite metric spaces
View Description Hide DescriptionA group G of symmetry transformations of the rays of an indefinite metric spaceV with metric operator η leads to a projective representation U of G in V in terms of η‐unitary, η‐antiunitary, η‐pseudounitary, and η‐pseudoantiunitary operators. We investigate the restrictions which are put on the irreducible components of U by the metric, and examine to what extent it is possible to decompose V into a direct sum of indefinite metric spaces, each carrying a projective representation of G. Attention is restricted to the cases where the subgroup of G which is represented by η‐unitary operators is of index 1 or 2.

Analytic expressions for the matrix elements of generators of Sp(6) in an Sp(6)⊇U(3) basis
View Description Hide DescriptionWork done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group Sp(6) in an Sp(6)⊇U(3) basis. Rosensteel has derived recursion relations for these matrix elements while Filippov has determined them using generating function technique, but it would also be convenient to have explicit and analytic formulas for them. This is what we do in this paper for the case of closed shells, i.e., when the irreducible representation (irrep) of Sp(6) is characterized by equal values for the three weight generators in the lowest weight state. We also indicate how our results can be extended to the case of arbitrary irreps of Sp(6), i.e., when we have open shells.

A pair of commuting missing label operators for G⊇[SU(2)]^{ n }
View Description Hide DescriptionA general procedure, which can lead to a pair of commuting subgroup scalars in the enveloping algebra of a Lie groupG, decomposed in [SU(2)]^{ n }, is discussed. The technique is illustrated by means of three explicit examples.

Structure constants for Lie algebras
View Description Hide DescriptionAn orthogonal basis in root space, related to the weights of the smallest representation, is used to provide a list of the algebraic conditions which the structure constants N _{αβ} must satisfy for all simple Lie algebras. A particular explicit set of solutions for all the N _{αβ} is given.

The semisimple subalgebras of Lie algebras
View Description Hide DescriptionTechniques for obtaining generators of nonregular maximal subalgebras of Lie algebras, alternative to those of Dynkin, are developed. They exploit the use of orthogonal bases in weight space, which are related to quark weights. The projection from an algebraG to its nonregular subalgebras g is related to an orthogonal matrix. The roots of G which project onto roots of g can be simply specified in the orthogonal bases. The phases of the expansion coefficients of generators of g in terms of generators of G are specified in such a manner that, for a given G, the entire set { g} of maximal subalgebras have consistent phases. The condition e _{−β} =e ^{°} _{β} is satisfied for all generators of g. The generators of all maximal nonregular subalgebras of all exceptional algebras are exhibited.

A unified description of the representations of the graded Lie algebra Gsl(2)
View Description Hide DescriptionIndecomposable representations of the graded Lie algebra Gsl(2) are analyzed in detail. It is further shown that the study of the irreducible representations (finite‐ and infinite‐dimensional) is intimately related to the study of these indecomposable representations.

On the Cauchy problem for the coupled Schrödinger–Klein–Gordon equations in one space dimension
View Description Hide DescriptionThe existence, uniqueness, and continuity with respect to initial data of global solutions of the Cauchy problem is proved for the Schrödinger and Klein–Gordon equations with Yukawa coupling in one space dimension. The proof is based on the standard tools for handling abstract nonlinear waveequations.

Sum rule for products of Bessel functions: Comments on a paper by Newberger
View Description Hide DescriptionRecently, Newberger considered a series of Bessel functions with as a special case the form ∑( n ^{ j } J ^{2} _{ n }(z))/(n+μ). The interesting point is that he obtained new explicit expressions for the sum of the series. In this note we point out that some results of Newberger are not correct, especially the results obtained by the principle of analytic continuation. Our remarks include a correction for his important result for the series ∑J _{ n }(z)J _{ n−m }(z)/(n+μ).

Fractional approximations for the spherically symmetric Coulomb scattering wave functions
View Description Hide DescriptionAn improved method is presented for obtaining fractional approximations. The fractional parameters are now solutions of a set of linear equations, and no nonlinear equations are involved as in the previous procedure. Excellent fractional approximations are presented for the Coulomb functions for η=0.5, 1, 2, and 5. The accuracy is sufficient for most of the computations where this function is used. The straightforward extension to higher orders is indicated.

Projection operator techniques in physics
View Description Hide DescriptionA systematic account of projection operators (projectors) and orthogonalization techniques together with applications to selected areas of physics is presented. This unified approach is shown to have advantages over other approaches in that the mathematical statements are more precise. The mathematical level, however, is aimed at the practicing physicist and lies between rigorous mathematics and current use in physics. Further, the techniques presented have practical applications as is demonstrated by examples in the quantum theory of measurement, in the relationship between second quantization and configuration space techniques, and in an account of generalized Wannier and Bloch functions. Attention is paid to the problem of construction of orthogonal projection operators (orthogonal projectors). The construction of orthogonal projectors even in approximate form would allow the solution of many practical problems ranging from the eigenvalue spectrum problem to the construction of states for many‐body systems. One can almost say that a n y problem in quantum mechanics can be formulated as a problem involving the construction of projectors.

Fractals and nonstandard analysis
View Description Hide DescriptionWe describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future.

Mean curvature and radiation field in SO(2) electrodynamics
View Description Hide DescriptionThe trivial bundle of orthonormal frames over flat space‐time is decomposed into two subbundles with structure groups SO(1,1) and SO(2), respectively. The curvature in the SO(2) bundle is identified with the electromagnetic field. It is shown that on certain conditions imposed upon the bundle decomposition the exterior derivative of the mean curvature 1‐form in the SO(2) bundle is equal to the curvature 2‐form in the SO(1,1) bundle. These conditions are (i) the Frobenius integrability of certain distributions generated by the splitting of the associated tangent bundle, and (ii) the vanishing of the mean curvature in the SO(1,1) bundle. For a single point charge the curvature in the SO(1,1) subbundle is identical to the radiation part of the Liénard–Wiechert field.

Equivalent random force and time‐series model in systems far from equilibrium
View Description Hide DescriptionUnder the condition that observed time‐series data is given, a stochastic Markovian equation for a physical system can be transformed into an observable non‐Markovian equation used in the time‐series analysis. The physical random force satisfying the fluctuation dissipation theorem is also transformed into a stochastically equivalent random force in the derivation of the time‐series model of observable variables. Statistical quantities, i.e., correlation and power spectral density functions for observable variables, can be expressed not only by the physical random force, but also by the equivalent random force. A relation between the variance of physical random force and that of equivalent random force is also found.

Maupertuis’ principle of least action in stochastic calculus of variations
View Description Hide DescriptionWithin the framework of stochastic calculus of variations for time‐symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time‐symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.

A bounded convergence theorem for the Feynman integral
View Description Hide DescriptionWe give a bounded convergence theorem for the Feynman integral for the class of bounded, measurable potentials.