Volume 27, Issue 8, August 1986
Index of content:

Group‐theoretic determination of normal coordinates for molecular vibration
View Description Hide DescriptionIt is shown that the normal coordinates for molecular vibration are determined by simply taking those linear combinations of mass‐weighted displacements that are irreducible bases of the group relevant to the molecule.

The state labeling problem—A universal solution
View Description Hide DescriptionA new method is proposed for labeling a basis for any irreducible representation, with top weight, of a simple Lie algebra (LA). It has two advantages over Gel’fand–Zetlin patterns: (i) being exactly the same for all LA’s both exceptional and classical and (ii) providing labels that are much more compact. Thus it may prove useful in discussing large representations such as those E _{8}⊗E _{8} that occur in superstring theory. The method makes essential use of the Weyl group and is based on a theorem that associates to each weight a symmetric matrix with integer coefficients whose rank equals the multiplicity of the weight.

Multiplicity‐free Wigner coefficients for semisimple Lie groups. I. The U(n) pattern calculus
View Description Hide DescriptionThis is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity‐free reduced Wigner coefficients for the Lie groups U(n) and O(n). The approach employed enables a direct (nonrecursive) derivation of reduced Wigner coefficients. The absolute value squared of the reduced Wigner coefficients is expressed as a rational polynomial function (numerator polynomial divided by denominator polynomial) whose Weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by Casimir invariants. A unified treatment of the composition laws of reduced Wigner coefficients and the evaluation of their associated denominator polynomials is presented, which applies to both U(n) and O(n). An explicit formula for the numerator polynomials of U(n) is also derived. The numerator polynomials for the orthogonal groups will be given in the second paper of the series.

Multiplicity‐free Wigner coefficients for semisimple Lie groups. II. A pattern calculus for O(n)
View Description Hide DescriptionIn this paper a direct, algebraic derivation of a large class of (multiplicity‐free) reduced Wigner coefficients is presented for the orthogonal groups. In particular all elementary reduced Wigner coefficients, including those for the fundamental spinor and tensor representations, are obtained. The results are presented in a form directly analogous to the corresponding results obtained for the unitary groups.

Superunitary reduction of the orthosymplectic supertableaux
View Description Hide DescriptionThe supertableaux of the orthosymplectic groups OSP(2ν‖2p) and OSP(2ν+1‖2p) are decomposed in sums of supertableaux of the superunitary group SU(ν‖p).

Closed, analytic, boson realizations for Sp(4)
View Description Hide DescriptionThe problem of determing a boson realization for an arbitrary irrep of the unitary simplectic algebra Sp(2d) [or of the corresponding discrete unitary irreps of the unbounded algebra Sp(2d,R)] has been solved completely in recent papers by Deenen and Quesne [J. Deenen and C. Quesne, J. Math. Phys. 2 3, 878, 2004 (1982); 2 5, 1638 (1984); 2 6, 2705 (1985)] and by Moshinsky and co‐workers [O. Castaños, E. Chacón, M. Moshinsky, and C. Quesne, J. Math. Phys. 2 6, 2107 (1985); M. Moshinsky, ‘‘Boson realization of symplectic algebras,’’ to be published]. This solution is not known in closed analytic form except for d=1 and for special classes of irreps for d>1. A different method of obtaining a boson realization that solves the full problem for Sp(4) is described. The method utilizes the chain Sp(2d)⊇SU(2)×SU(2) ×⋅⋅⋅×SU(2) (d times), which, for d≥4, does not provide a complete set of quantum numbers. Though a simple solution of the missing label problem can be given, this solution does not help in the construction of a mapping algorithm for general d.

Finite‐dimensional irreducible representations of the Lie superalgebra sl(1,3) in a Gel’fand–Zetlin basis
View Description Hide DescriptionA concept of a Gel’fand–Zetlin pattern for the Lie superalgebra sl(1,3) is introduced. Within every finite‐dimensional irreducible sl(1,3) module the set of the Gel’fand–Zetlin patterns constitute an orthonormed basis, called a Gel’fand–Zetlin basis. Expressions for the transformation of this basis under the action of the generators are written down for every finite‐dimensional irreducible representation.

The representation of Lie groups by bundle maps
View Description Hide DescriptionThis paper deals with representations of connected Lie groups by bundle maps of fiber bundles. It is pointed out that a large class of such representations can be obtained from the bundle structure theroem, and explicit constructions are given, first on principal bundles and then on associated bundles. Examples are provided to show that, for bundle representations, the theorem of full reducibility breaks down even for compact Lie groups. Finally, a general construction is given for obtaining representations of a Lie group on an arbitrary principal bundle. However, it is not known whether this exhausts all possibilities.

SU(m/n) weight systems and superprojection matrices
View Description Hide DescriptionRepresentations of the SU(m/n) superalgebra are studied in terms of the Kac–Dynkin weight systems. Superprojection matrices are introduced for the possible branching patterns.

The Hirota conditions
View Description Hide DescriptionThe condition on the polynomialP for a Hirota equation Pτ ⋅ τ=0 to have an N‐soliton solution for arbitrary N is examined and simplified.

Spinor structures on spheres and projective spaces
View Description Hide DescriptionAn explicit construction of spinor structures on real, complex, and quaternionic projective spaces is given for all cases when they exist. The construction is based on a theorem describing the bundle of orthonormal frames of a homogeneous Riemannian manifold. This research is motivated by a remarkable coincidence of spinor connections on low‐dimensional spheres with simple, topologically nontrivial gauge configurations.

Spontaneous compactification and Ricci‐flat manifolds with torsion
View Description Hide DescriptionThe Freund–Rubin mechanism is based on the equation R _{ i k }=λg _{ i k } (where λ>0), which, via Myers’ theorem, implies ‘‘spontaneous’’ compactification. The difficulties connected with the cosmological constant in this approach can be resolved if torsion is introduced and λ is set equal to zero, but then compactification ‘‘by hand’’ is necessary since the equation R _{ i k } =0 can be satisfied both on compact and on noncompact manifolds. In this paper we discuss the global geometry of Ricci‐flat manifolds with torsion, and suggest ways of restoring the ‘‘spontaneity’’ of the compactification.

Transverse nearest neighbor degeneracies on a 2×N lattice
View Description Hide DescriptionAn exact 15‐term recursion relation and the associated generating function are derived for the number of arrangements of q particles on a 2×N lattice such that s occupied nearest neighbor pairs, v of which are transverse, and t unoccupied nearest neighbor pairs are formed.

Special solutions of the Sparling equation
View Description Hide DescriptionThe Sparling equation, a first‐order, matrix‐valued linear differential equation that is equivalent to the self‐dual Yang–Mills equations for any group, has recently been solved by quadratures for the case of SL(2,C) or its subgroup. It is the purpose of this paper to show how for a series of special cases, rather than integrating the quadratures, the Sparling equation can be reduced to an algebraic equation and then solved, yielding the single‐ and multi‐instanton fields parallel to isospace.

Nonlinear classical scalar field theory in curved space‐time
View Description Hide DescriptionA generic Lagrangian based classical field theory is formulated for any space‐time manifold in which certain postulated conditions remain valid. The choice of a specific field Lagrangian leads to a nonlinear modeltheory that admits a rigorous closed‐form particlelike solution in isotropic homogeneous space‐time of positive spatial curvature. This metastable solution, of finite positive energy, is discussed in relation to its counterpart in isotropic homogeneous space‐time of negative spatial curvature.

Wigner quantum systems. Two particles interacting via a harmonic potential. I. Two‐dimensional space
View Description Hide DescriptionA noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center‐of‐mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite‐dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different values. For any non‐negative integer or half‐integer l there exists a representation, where the orbital momentum is l (in units of 2ℏ). The position of any one of the particles cannot be localized, since the operators of the coordinates do not commute with each other. The constituents are smeared with a certain probability within a finite surface, which moves with a constant velocity together with the center of mass.

Continuity of entropy and mutual entropy in C*‐dynamical systems
View Description Hide DescriptionThe lower semicontinuities of the entropy and the mutual entropy of a state for C*‐dynamical systems are proved with respect to the set of all KMS states and the set of all α‐invariant states.

Green’s function for motion in Coulomb‐modified separable nonlocal potentials
View Description Hide DescriptionA closed form expression is derived for the outgoing wave radial Green’s functionG^{(+)} _{ l } (r,r’) for motion in the Coulomb plus rank one separable nonlocal potential with form factorv _{ l }(r)=2^{−l } ×(l!)^{−} ^{1} r ^{ l } e ^{−β l r }. Some possible applications of the result are discussed.

Conformally symmetric radiating spheres in general relativity
View Description Hide DescriptionA method used to study the evolution of radiating anisotropic (principal stresses unequal) spheres is applied to the case in which the space‐time (within the sphere) admits a one‐parameter group of conformal motions. Two different kind of models are obtained, depending on the equation of state for the stresses. In one case the energy flux density at the boundary of the sphere (the luminosity) should be given as a function of the timelike coordinate in order to integrate the system of equations. In the other case the luminosity is inferred from the equation of state for the stresses. Both models are integrated numerically and their eventual applications to some astrophysical problems are discussed.

Dual mass, H‐spaces, self‐dual gauge connections, and nonlinear gravitons with topological origin
View Description Hide DescriptionAn analogy between source‐free, asymptotically Taub–NUT magnetic monopolesolutions to Einstein’s equation and self‐(anti‐self‐) dual gauge connections is displayed, which finds its origin in the first Chern class of these space‐times. A definition of asymptotic graviton modes is proposed that suggests that a subclass of Penrose’s nonlinear gravitons or Newman’s H‐spaces could emerge from nontrivial space‐time topologies.