Volume 26, Issue 12, December 1985
Index of content:

Nonlinear spinor representations
View Description Hide DescriptionA systematic method for the construction of nonlinear carrier spaces for a class of nonlinear spinor representations of complex and pseudo‐orthogonal rotation groups is presented. It is shown that Cartan pure spinors, which satisfy quadratic constraints, are special cases of our construction. A class of new nonlinear spinor representations is discovered, which is particularly interesting in the case of generalized conformal groups SO(ν+1,ν−1), ν=3,4,... . The nonlinearity condition considerably diminishes the number of independent spinor components and therefore the corresponding spinor fields are the most natural building blocks for grand unified field theories. The method presented here is universal and can be applied for the construction of new nonlinear representations of other higher‐symmetry groups.

On the projective representations of finite Abelian groups. II
View Description Hide DescriptionComplete sets of inequivalent irreducible projective representations of C ^{ n } _{ m } ={w _{1},⋅⋅⋅,w _{ n }; w ^{ m } _{ i } =1, i=1,...,n; w _{ i } w _{ j } =w _{ j } w _{ i }, i, j=1,...,n} with respect to a class of factor sets α are determined, where α(w _{ i },w _{ j }) =θα(w _{ j },w _{ i }), 1≤i<j≤n and θ is a fixed mth root of unity. A single irreducible projective representation of C ^{ n } _{ m } for each factor set α is constructed and called the basic projective representation. The rest of the representations are obtained by tensoring the basic projective representations with some ordinary representations of C ^{ n } _{ m }. Projective representations of C ^{ n } _{ m } are thus parametrized in terms of its ordinary representations.

The group of gauge transformations as a Schwartz–Lie group
View Description Hide DescriptionThe group of gauge transformations of a smooth principal bundle P(M,G) over a not necessarily compact manifoldM and with a not necessarily compact structure group G is proved to be a Schwartz–Lie group. Its Lie algebra and exponential map are discussed.

Lie symmetries of some equations of the Fokker–Planck type
View Description Hide DescriptionThe structure of the local Lie groups of symmetries of some partial differential equations of the Fokker–Planck type in one space dimension is investigated. A connection between these groups and the group SL_{2}(R) is established in the sense that they are all shown to be locally isomorphic to SL_{2}(R)A, where A is the radical. It is conjectured that the groups of Lie symmetries of all Fokker–Planck equations in one space dimension have this structure. The notion of partial invariance, due to Ovsiannikov, is applied to the equations studied. It appears plausible that the class of partially invariant solutions of these equations is larger than the class of invariant solutions although no explicit demonstration of this claim is available at present.

The infinite coupling limit of perturbative expansions from a variational extrapolation method
View Description Hide DescriptionA method for extrapolating perturbative power series to infinity is described. It is a Borel partial resummation stabilized by a variational parameter. Two kinds of series relative to the anharmonic oscillators ‖x‖^{ k }, k>0, are extrapolated in order to illustrate the effectiveness of the method: the Rayleigh–Schrödinger series, on the one hand, which, after extrapolation, provides the strong coupling expansion of the energy levels, and their lattice expansion, on the other hand, from which is extracted the continuum limit.

Dynamical group chains and integrity bases
View Description Hide DescriptionAn algorithm for constructing a Hamiltonian from the generators of a dynamical group G, which is invariant under the operations of a symmetry group H ⊆ G, is presented. In practice, this algorithm is subject to a large number of simplifications. It is sufficient to construct an integrity basis of H scalars in terms of which all H scalars can be expressed as polynomial functions. In many instances the integrity basis exists in 1–1 correspondence with the Casimir operators for a group–subgroup lattice based on the pair H ⊆ G. When this is so the theory embodies natural symmetry limits and analytic results for observables can be given. Examples of the application of the algorithm are given for the dynamical group SU(2) with symmetry subgroups C _{3} and U(1) and for SU(N) ⊇ SO(3), N=3, 4, and 6.

The Green’s function for a finite linear chain
View Description Hide DescriptionA new expression for the Green’s function of a finite‐length one‐dimensional harmonic lattice with nearest‐neighbor interactions is reported. Simple closed expressions in terms of Chebyshev polynomials are developed for periodic, fixed, and free end boundary conditions.

Integrable Hamiltonian systems with velocity‐dependent potentials
View Description Hide DescriptionThe integrability of a two‐dimensional Hamiltonian in which the potential depends explicitly on the momenta is investigated. Hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field. Two integrable classes of potentials are identified and the second integral of motion is constructed for each of them. The singularity analysis of the equations of motion is also performed, confirming once more the relation between the (weak) Painlevé property and integrability.

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity‐dependent mappings of second‐order differential equations
View Description Hide DescriptionThe primary purpose of this paper is to show that infinitesimal velocity‐dependent symmetry mappings [(a) x̄^{ i } =x ^{ i } +δx ^{ i }, δx ^{ i } ≡ ξ^{ i }(ẋ,x,t)δa with associated change in path parameter (b) t̄=t+δt, δt ≡ ξ^{0}(ẋ,x,t)] of classical (including relativistic) particle systems (c) E ^{ i }(ẍ,ẋ,x,t) =0 are expressible in a form with a characteristic functional structure which is the same for all dynamical systems (c) and is manifestly dependent upon constants of motion of the system. In this characteristic form the symmetry mappings are determined by (d) ξ^{ i } =Z ^{ i }(ẋ,x,t) +ẋ^{ i }ξ^{0},ξ^{0} arbitrary; the functions Z ^{ i } appearing in (d) have the form (e) Z ^{ i } =B ^{ A } g ^{ i } _{ A }(C ^{1},...,C ^{ r }; t), 0≤r≤2n, A=1,...,2n, where the B ^{ A } are arbitrary constants of motion and the C’s appearing in the functions g ^{ i } _{ A } are specified constants of motion.
A procedure is given to determine the g ^{ i } _{ A }. For Lagrangian systems it follows that velocity‐dependent Noether mappings are a subclass of the above‐mentioned general symmetry mappings of the form (a)–(e). An analysis of velocity‐dependent Noether mapping theory is included in order to compare for Lagrangian systems the procedure for obtaining the characteristic form (e) of the general mappings with the procedure for obtaining the well‐known formula (f) Z ^{ i } _{ N } =H ^{ i j }(ẋ,x,t)∂Z/∂ẋ^{ j } (Z ≡ constant of motion), characteristic of velocity‐dependent Noether mappings. It is shown how any g i v e n velocity‐dependent symmetry mapping function Z ^{ i }(ẋ,x,t) (including Noether mappings) can be expressed in the form (e). A collection of variational formulas and identities is derived in order to develop from first principles the velocity‐dependent symmetry mapping theory. Throughout, comparisons are made between velocity‐dependent and velocity‐independent symmetry theory.

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. II. Mappings of first‐order differential equations
View Description Hide DescriptionInfinitesimal velocity‐dependent symmetry mappings of second‐order dynamical systems (a) E ^{ i }(ẍ, ẋ, x, t)≡ẍ^{ i } −F ^{ i }(ẋ, x, t)=0, i=1,..., n, were studied in considerable detail in a previous paper [J. Math. Phys. 2 X, xxxx (198X), the first of this series]. Among the results developed in that paper was a procedure for determining the characteristic functional structure of symmetry mappings for such second‐order systems. In this present companion paper it is shown that a similar procedure may be used to obtain the characteristic functional structure of infinitesimal symmetry mappings (b) ȳ^{ I }=y ^{ I }+δy ^{ I }, δy ^{ I } ≡η^{ I }(y, t)δa; (c) t̄=t+δt, δt≡η^{0}(y, t)δa, for systems of first‐order differential equations (d) E ^{ I }(ẏ, y, t)≡ẏ^{ I }−λ^{ I } (y, t)=0, I=1,..., N. This characteristic structure is the same for all first‐order systems (d) and is explicitly dependent upon constants of motion of the system. For the special case in which (d) is a system of N=2n equations derived from a system of n second‐order equations (a) it is shown how the respective symmetry equations based upon these two equivalent dynamical descriptions are related and how their symmetry solutions are correlated. Two examples are given.

Exact reduced density matrices for a model problem
View Description Hide DescriptionThe reduced density matrices of arbitrary order for the boson problem of N particles, each attracted harmonically to a central point and interacting with each other harmonically, are analytically calculated.

On the spectra of SO(3) scalars in the enveloping algebra of SU(3)
View Description Hide DescriptionFormulas are given that make it possible to calculate the eigenvalues of the two independent SO(3) scalars O ^{0} _{ l } and Q ^{0} _{ l } in the SU(3) enveloping algebra.

On Komar integrals in asymptotically anti‐de Sitter space‐times
View Description Hide DescriptionRecently, boundary conditions governing the asymptotic behavior of the gravitational field in the presence of a negative cosmological constant have been introduced using Penrose’s conformal techniques. The subsequent analysis has led to expressions of conserved quantities (associated with asymptotic symmetries) involving asymptotic Weyl curvature. On the other hand, if the underlying space‐time is equipped with isometries, a generalization of the Komar integral which incorporates the cosmological constant is also available. Thus, in the presence of an isometry, one is faced with two apparently unrelated definitions. It is shown that these definitions agree. This coherence supports the choice of boundary conditions for asymptotically anti‐de Sitter space‐times and reinforces the definitions of conserved quantities.

Some exact inhomogeneous solutions of Einstein’s equations with symmetries on the hypersurfaces t=const
View Description Hide DescriptionThe solution of Einstein’s field equations is studied for a metric written in the form (δ≠γ)d s ^{2}=−α^{2}(t,r,θ,φ)d t ^{2} +e ^{2β(t,r)} d r ^{2}+e ^{2γ(t,r)} dθ^{2} +e ^{2δ(t,r)} M ^{2}(θ)dφ^{2}. A perfect fluid, which flows orthogonally to the hypersurfaces t=const is considered as matter content. These hypersurfaces admit a translational Killing vector, which will not be, in general, a Killing vector of the whole space‐time. All the possible solutions are obtained when α depends on the variable φ. These solutions represent either a perfect fluid without expansion or vacuum with a cosmological constant Λ0. Some particular inhomogeneous solutions are obtained for α independently of φ. These solutions are physical, the fluid obeys an equation of statep=ρ (stiff matter), and the space‐time admits, apparently, only a group G _{2} of isometries. A vacuum family is also obtained in this case.

Anisotropic cosmological model in Nordtvedt’s scalar–tensor theory of gravitation
View Description Hide DescriptionAnisotropic cosmological models are considered in the light of the scalar–tensor theory of gravitation proposed by Nordtvedt. Special attention is paid to Bianchi type I models. The models consist of perfect fluid with the equation of statep=ερ. The solutions are obtained in Dicke’s conformally transformed units for empty space, as well as for ε=1 and (1)/(3) , assuming two separate functional relationships between ω and φ. Their properties are also compared with those of the models given in Brans–Dicke theory.

An extension of quaternionic metrics to octonions
View Description Hide DescriptionA treatment of a non‐Riemannian geometry including internal complex, quaternionic, and octonionic space is made. Then, an interpretation of this geometry for the nonsymmetric theory of Einstein–Schrödinger, and for the unified theory of Borchsenius is showed. Finally, field equations in the extended octonionic geometry of space‐time are obtained through a minimal action principle.

Two‐mode para‐Bose coherent states
View Description Hide DescriptionThe construction of eigenstates of the square of annihilation operators for a two‐mode para‐Bose system is reported. Bose coherent states can be deduced from these eigenstates as a special case. These states are termed para‐Bose coherent states. These states are degenerate. The expansion of the coherent states in terms of two‐mode para‐Bose energy eigenstates has been obtained and their salient properties are discussed. Also discussed is the uncertainty product of the square of position and momentum operators in the para‐Bose coherent states for a two‐mode system.

On the energy–time conjugation in quantum physics
View Description Hide DescriptionThe energy–time conjugation is discussed in terms of mutually Laplace conjugated positive variables. The quantum statistical distribution functions in the energy are thus put into correspondence with ‘‘draw’’ distributions in the time, represented by sums of Dirac δ distributions. Time averages on correlation functions correspond to ensemble averages in energy. Conversely energy coupling of systems can be represented by a special operation on the δ distributions in time. For this aim, the c o n n e c t i o n of distributions is introduced, which enables one to take into account in some ‘‘multiplicative’’ way their simultaneous and cooperative effects. The Appendix is entirely devoted to the definition and properties of these connections and to some aspects of their algebra that make them suitable for treating some fundamental problems bound to the necessity of accounting interactive effects of singularities.

Dirac quantization of a three‐dimensional gauge theory
View Description Hide DescriptionA model recently proposed by Hagen is examined from the point of view of Dirac quantization of constrained systems. This model exhibits interesting particular features for the Dirac method itself. Among them are the odd number of second‐class constraints and the fact that, when a gauge is fixed, constraints result from compatibility conditions between Lagrange multipliers. From the point of view of the model itself, the invalidity of the axial gauge in the non‐Abelian case is obtained by comparing the effective Hamiltonians for two different values of the arbitrary spacelike vector.

Color analysis, theory of Γ‐graded integrable evolution equations, and super Nijenhuis operators
View Description Hide DescriptionUsing the generalized Grassmann variables or color variables, a theory of Γ‐graded integrable evolution equations is presented by elevating the treatments of Magri, Gel’fand–Dorfman, and Fuchssteiner of nonlinear integrable bosonic evolution equations to the Γ‐graded case. As an example, it is shown that Kupershmidt’s super‐KdV is characterized by a Z_{2}‐graded Nijenhuis operator compatible with the underlying Z_{2}‐graded Hamiltonian structure.