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Dec 1985

Volume 26, Issue 12, pp. 3021-3204

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Nonlinear spinor representations

P. Furlan and R. Raczka

J. Math. Phys. 26, 3021 (1985); http://dx.doi.org/10.1063/1.526678 (12 pages) | Cited 5 times

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A 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.
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02.20.Rt Discrete subgroups of Lie groups
02.20.Qs General properties, structure, and representation of Lie groups

On the projective representations of finite Abelian groups. II

M. Saeed‐ul‐Islam

J. Math. Phys. 26, 3033 (1985); http://dx.doi.org/10.1063/1.526679 (3 pages) | Cited 1 time

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Complete sets of inequivalent irreducible projective representations of Cnm ={w1,⋅⋅⋅,wn; wmi =1, i=1,...,n; wiwj =wjwi, i, j=1,...,n} with respect to a class of factor sets α are determined, where α(wi,wj) =θα(wj,wi), 1≤i<j≤n and θ is a fixed mth root of unity. A single irreducible projective representation of Cnm 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 Cnm. Projective representations of Cnm are thus parametrized in terms of its ordinary representations.
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02.20.Bb General structures of groups

The group of gauge transformations as a Schwartz–Lie group

Renzo Cirelli and Alessandro Manià

J. Math. Phys. 26, 3036 (1985); http://dx.doi.org/10.1063/1.526680 (6 pages) | Cited 1 time

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The group of gauge transformations of a smooth principal bundle P(M,G) over a not necessarily compact manifold M 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.
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02.20.Tw Infinite-dimensional Lie groups
02.40.Vh Global analysis and analysis on manifolds

Lie symmetries of some equations of the Fokker–Planck type

C. C. A. Sastri and K. A. Dunn

J. Math. Phys. 26, 3042 (1985); http://dx.doi.org/10.1063/1.526681 (6 pages) | Cited 9 times

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The 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 SL2(R) is established in the sense that they are all shown to be locally isomorphic to SL2(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.
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02.30.Jr Partial differential equations
02.20.Sv Lie algebras of Lie groups
02.20.Rt Discrete subgroups of Lie groups

The infinite coupling limit of perturbative expansions from a variational extrapolation method

B. Bonnier, M. Hontebeyrie, and E. H. Ticembal

J. Math. Phys. 26, 3048 (1985); http://dx.doi.org/10.1063/1.526682 (5 pages) | Cited 7 times

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A 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.
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02.60.Ed Interpolation; curve fitting
02.30.Sa Functional analysis

Dynamical group chains and integrity bases

R. Gilmore and J. P. Draayer

J. Math. Phys. 26, 3053 (1985); http://dx.doi.org/10.1063/1.526683 (15 pages) | Cited 20 times

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An 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 C3 and U(1) and for SU(N) ⊇ SO(3), N=3, 4, and 6.
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02.20.Rt Discrete subgroups of Lie groups
11.30.-j Symmetry and conservation laws
21.60.Fw Models based on group theory
31.15.-p Calculations and mathematical techniques in atomic and molecular physics

The Green’s function for a finite linear chain

Ronald Bass

J. Math. Phys. 26, 3068 (1985); http://dx.doi.org/10.1063/1.526684 (2 pages) | Cited 2 times

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A 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.
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45.05.+x General theory of classical mechanics of discrete systems
63.20.D- Phonon states and bands, normal modes, and phonon dispersion

Integrable Hamiltonian systems with velocity‐dependent potentials

B. Dorizzi, B. Grammaticos, A. Ramani, and P. Winternitz

J. Math. Phys. 26, 3070 (1985); http://dx.doi.org/10.1063/1.526685 (10 pages) | Cited 34 times

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The 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.
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45.05.+x General theory of classical mechanics of discrete systems
02.20.Qs General properties, structure, and representation of Lie groups
02.30.Hq Ordinary differential equations
41.60.-m Radiation by moving charges

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity‐dependent mappings of second‐order differential equations

Gerald H. Katzin and Jack Levine

J. Math. Phys. 26, 3080 (1985); http://dx.doi.org/10.1063/1.526686 (20 pages) | Cited 7 times

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The primary purpose of this paper is to show that infinitesimal velocity‐dependent symmetry mappings [(a) mathi =xi +δxi, δxi  ≡ ξi(math,x,t)δa with associated change in path parameter (b) math=t+δt, δt ≡ ξ0(math,x,t)] of classical (including relativistic) particle systems (c) Ei(math,math,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) ξi =Zi(math,x,t) +mathiξ0,ξ0 arbitrary; the functions Zi appearing in (d) have the form (e) Zi =BAgiA(C1,...,Cr; t), 0≤r≤2n, A=1,...,2n, where the BA are arbitrary constants of motion and the C’s appearing in the functions giA are specified constants of motion.
A procedure is given to determine the giA. 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) ZiN =Hij(math,x,t)∂Z/∂math j (Z ≡  constant of motion), characteristic of velocity‐dependent Noether mappings. It is shown how any given velocity‐dependent symmetry mapping function Zi(math,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.
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45.05.+x General theory of classical mechanics of discrete systems
03.30.+p Special relativity
04.20.-q Classical general relativity
02.30.Jr Partial differential equations

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. II. Mappings of first‐order differential equations

Gerald H. Katzin and Jack Levine

J. Math. Phys. 26, 3100 (1985); http://dx.doi.org/10.1063/1.526687 (5 pages) | Cited 4 times

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Infinitesimal velocity‐dependent symmetry mappings of second‐order dynamical systems (a) Ei(math, math, x, t)≡mathi −Fi(math, x, t)=0, i=1,..., n, were studied in considerable detail in a previous paper [J. Math. Phys. 2X, 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) mathI=yI+δyI, δyI ≡ηI(y, t)δa; (c) math=t+δt, δt≡η0(y, t)δa, for systems of first‐order differential equations (d) EI(math, y, t)≡mathI−λI (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.
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45.05.+x General theory of classical mechanics of discrete systems
04.20.-q Classical general relativity
03.30.+p Special relativity
02.30.Jr Partial differential equations

Exact reduced density matrices for a model problem

Leon Cohen and Chongmoon Lee

J. Math. Phys. 26, 3105 (1985); http://dx.doi.org/10.1063/1.526688 (4 pages) | Cited 14 times

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The 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.
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03.65.Ge Solutions of wave equations: bound states
31.15.-p Calculations and mathematical techniques in atomic and molecular physics
03.65.Fd Algebraic methods

On the spectra of SO(3) scalars in the enveloping algebra of SU(3)

H. De Meyer, G. Vanden Berghe, and J. Van der Jeugt

J. Math. Phys. 26, 3109 (1985); http://dx.doi.org/10.1063/1.526689 (3 pages) | Cited 12 times

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Formulas are given that make it possible to calculate the eigenvalues of the two independent SO(3) scalars O0l and Q0l in the SU(3) enveloping algebra.
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03.65.Fd Algebraic methods
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Rt Discrete subgroups of Lie groups

On Komar integrals in asymptotically anti‐de Sitter space‐times

Anne Magnon

J. Math. Phys. 26, 3112 (1985); http://dx.doi.org/10.1063/1.526690 (6 pages) | Cited 9 times

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Recently, 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.
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04.20.-q Classical general relativity
02.40.Ky Riemannian geometries

Some exact inhomogeneous solutions of Einstein’s equations with symmetries on the hypersurfaces t=const

F. Argüeso and J. L. Sanz

J. Math. Phys. 26, 3118 (1985); http://dx.doi.org/10.1063/1.526691 (7 pages)

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The solution of Einstein’s field equations is studied for a metric written in the form (δ≠γ)ds2=−α2(t,r,θ,φ)dt2 +e2β(t,r) dr2+e2γ(t,r) dθ2 +e2δ(t,r)M2(θ)dφ2. 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 state p=ρ (stiff matter), and the space‐time admits, apparently, only a group G2 of isometries. A vacuum family is also obtained in this case.
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04.20.Jb Exact solutions

Anisotropic cosmological model in Nordtvedt’s scalar–tensor theory of gravitation

A. Banerjee, N. Banerjee, and N. O. Santos

J. Math. Phys. 26, 3125 (1985); http://dx.doi.org/10.1063/1.526692 (6 pages) | Cited 6 times

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Anisotropic 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 state p=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.
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04.50.-h Higher-dimensional gravity and other theories of gravity
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

An extension of quaternionic metrics to octonions

S. Marques and C. G. Oliveira

J. Math. Phys. 26, 3131 (1985); http://dx.doi.org/10.1063/1.526693 (9 pages) | Cited 8 times

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A 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.
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04.50.-h Higher-dimensional gravity and other theories of gravity
04.20.Cv Fundamental problems and general formalism
04.20.Fy Canonical formalism, Lagrangians, and variational principles

Two‐mode para‐Bose coherent states

G. M. Saxena and C. L. Mehta

J. Math. Phys. 26, 3140 (1985); http://dx.doi.org/10.1063/1.526694 (6 pages) | Cited 1 time

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The 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.
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05.30.-d Quantum statistical mechanics

On the energy–time conjugation in quantum physics

F. Bailly

J. Math. Phys. 26, 3146 (1985); http://dx.doi.org/10.1063/1.526695 (9 pages)

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The 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 connection 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.
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05.30.-d Quantum statistical mechanics
02.30.Uu Integral transforms
02.30.Vv Operational calculus
02.50.-r Probability theory, stochastic processes, and statistics

Dirac quantization of a three‐dimensional gauge theory

A. Burnel and M. Van Der Rest‐Jaspers

J. Math. Phys. 26, 3155 (1985); http://dx.doi.org/10.1063/1.526696 (5 pages) | Cited 2 times

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A 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.
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11.10.Ef Lagrangian and Hamiltonian approach
11.15.-q Gauge field theories

Color analysis, theory of Γ‐graded integrable evolution equations, and super Nijenhuis operators

Robert Trostel

J. Math. Phys. 26, 3160 (1985); http://dx.doi.org/10.1063/1.526643 (12 pages) | Cited 1 time

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Using 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 Z2‐graded Nijenhuis operator compatible with the underlying Z2‐graded Hamiltonian structure.
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11.10.Ef Lagrangian and Hamiltonian approach
11.30.Pb Supersymmetry
02.20.Qs General properties, structure, and representation of Lie groups
02.40.Hw Classical differential geometry

Spectrum doubling and double‐valuedness

Richard Stacey

J. Math. Phys. 26, 3172 (1985); http://dx.doi.org/10.1063/1.526644 (4 pages) | Cited 1 time

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The effect of making the lattice Dirac operator the square root of the Laplacian is investigated. The doubling of the fermion spectrum is then matched by that of the boson spectrum, unless bosons are restricted to a double‐spaced sublattice. Fermion spectrum doubling is found to be a necessary consequence of the double‐valuedness of the square root.
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11.15.Ha Lattice gauge theory

Differential characters: The Dirac monopole as an example

Robert Coquereaux

J. Math. Phys. 26, 3176 (1985); http://dx.doi.org/10.1063/1.526645 (4 pages) | Cited 1 time

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Choosing the Dirac monopole as an example, the theory of differential characters is sketched and the quantization condition is recovered in a new way without considering singularities and without using a global formulation of gauge fields (i.e., without using fiber bundle techniques).
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11.15.-q Gauge field theories
02.40.Sf Manifolds and cell complexes
02.40.Hw Classical differential geometry

Spinorial infinite equations fitting metric‐affine gravity

A. Cant and Y. Ne’eman

J. Math. Phys. 26, 3180 (1985); http://dx.doi.org/10.1063/1.526646 (10 pages) | Cited 12 times

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Two different approaches are used to construct infinite‐component spinor equations based on the multiplicity‐free irreducible representations of SL(4,R). These ‘‘manifield’’ equations are SL(2,C) invariant; they exist in special relativity, and can directly be coupled to gravitation in the metric‐affine theory, i.e., in Einstein’s general relativity with nonpropagating torsion and nonmetricity. In the first approach the maximal compact subgroup SŌ(4) of SL(4,R) is ‘‘physical.’’ A vector operator X μ is constructed directly in the infinite‐dimensional reducible representation Ddisc( 1/2 ,0) ⊕Ddisc(0, 1/2 ). In the second approach, SL(2,C) and a vector operator γ μ are embedded directly in SL(4,R) via the Dirac representation. A manifield equation is then constructed (in a manner analogous to the Majorana equation) by taking an infinite‐dimensional irreducible multiplicity‐free representation of SL(4,R), spinorial in  j1, in the ( j1, j2) reduction over SŌ(4). Both manifields can fit the observed mass spectrum.
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11.90.+t Other topics in general theory of fields and particles (restricted to new topics in section 11)
04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Rt Discrete subgroups of Lie groups

Fluctuation–dissipation theorem for QCD plasma

J. Chakrabarti

J. Math. Phys. 26, 3190 (1985); http://dx.doi.org/10.1063/1.526647 (3 pages)

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We explore a quantum‐chromodynamic (QCD) plasma in stationary nonequilibrium states assuming that the process of thermalization is governed by Fokker–Planck dynamics. The generalized thermodynamic potential appropriate to the state is obtained. A relationship is developed between the response function and the fluctuations in the stationary state.
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12.38.Aw General properties of QCD (dynamics, confinement, etc.)

Simple calculation of Löwdin’s alpha function. II. Easier procedure for evaluating bK k(L M‖l ), and vanishing of hn,2n−i(L M‖l ) for special values of i and n

Noboru Suzuki

J. Math. Phys. 26, 3193 (1985); http://dx.doi.org/10.1063/1.526648 (7 pages) | Cited 10 times

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This paper subsequent to the one [J. Math. Phys. 25, 1133 (1984)] (referred to as Part I) presents the following new results: It is found out that for M=L and L−1 the coefficients bK k(L M‖l) in Löwdin’s α‐function have properties other than manifested in Part I. The expression for bK k(L M‖l) is shown to be equivalent to the one into which Sharma’s expression, obtained in a different manner from that in Part I, is simplified by Rashid. The use of Rashid’s expression leads to the recurrence formula for bK k(L M‖l) with respect to M only. This formula and the expression for the bK k(L M‖l) with M=L provide an easier procedure for successively evaluating bK k(L M‖l) than in Part I. Furthermore, it is proved that the coefficients hn,2n−i(L M‖l) in the asymptotic form of the α‐function vanish for i<l+M and for n<l.
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31.15.-p Calculations and mathematical techniques in atomic and molecular physics
71.10.-w Theories and models of many-electron systems
02.30.Mv Approximations and expansions
02.30.-f Function theory, analysis
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