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Dec 1986

Volume 27, Issue 12, pp. 2823-3072

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Irreducible ∗‐representations of Lie superalgebras B(0,n) with finite‐degenerated vacuum

J. Blank and M. Havlíček

J. Math. Phys. 27, 2823 (1986); http://dx.doi.org/10.1063/1.527257 (9 pages) | Cited 4 times

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The problem of getting irreducible ∗‐representations π of Lie superalgebras B(0,n), n=1,2, is studied, starting with a recently constructed family of linear representations in terms of differential operators on the space C∞N of CN ‐valued C∞ ‐functions. Equivalent formulation via creation‐annihilation operators of a para‐Bose system with n degrees of freedom is used, and the domain D of any π is shown to be a subset of C∞N containing a nonzero vacuum subspace. By assuming its dimension finite, the necessary conditions for existence of π are derived. The method is applied to the superalgebra B(0,1) and a one‐parameter family Π of nonequivalent irreducible ∗‐representations in terms of unbounded linear operators on L2(R+)⊗C2 is obtained. Each representation π∊Π has a nondegenerated vacuum and for all z∊B(0,1) satisfying z=z∗, the operators π(z) are essentially self‐adjoint.
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02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups
11.30.Pb Supersymmetry

The generalized atypical supertableaux of the orthosymplectic groups OSP(2‖2p)

Michel Gourdin

J. Math. Phys. 27, 2832 (1986); http://dx.doi.org/10.1063/1.527258 (10 pages) | Cited 1 time

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The classification and the interpretation of the Young supertableaux of the orthosymplectic group OSP(2‖2p) are given. A particular emphasis is made on the generalized atypical supertableaux associated to nonfully reducible atypical representations.
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02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups
11.30.Pb Supersymmetry

Principal five‐dimensional subalgebras of Lie superalgebras

Joris Van der Jeugt

J. Math. Phys. 27, 2842 (1986); http://dx.doi.org/10.1063/1.527259 (6 pages) | Cited 2 times

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The analog of sl(2) for Lie superalgebras is osp(1,2), a five‐dimensional superalgebra. All basic classical Lie superalgebras L that contain a principal five‐dimensional osp(1,2) subalgebra are classified. Moreover, the decomposition of the standard representation and of the adjoint representation of L into irreducible components of the principal osp(1,2) subalgebra is given.
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02.20.Sv Lie algebras of Lie groups
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Rt Discrete subgroups of Lie groups

Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable?

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

J. Math. Phys. 27, 2848 (1986); http://dx.doi.org/10.1063/1.527260 (5 pages) | Cited 44 times

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The Kadomtsev–Petviashvili (KP) hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy. Since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the KP hierarchy provides integrable scalar equations with an arbitrary number of independent variables. It is shown, by investigating a specific equation in 3+1 dimensions, that the higher equations in the KP hierarchy are only integrable in a conditional sense. The equation under study, taken in isolation, does not pass certain well‐known and reliable integrability tests. Thus, applying Painlevé analysis, we find that solutions exist, allowing movable critical points. Furthermore, solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions. On the other hand, if the dependence of a solution on the first 2+1 variables is restricted by the fact that it should also satisfy the KP equation itself, then the integrability conditions in the other dimensions are satisfied. ‘‘Conditional integrability’’ thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy.
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02.30.Jr Partial differential equations
45.05.+x General theory of classical mechanics of discrete systems

Prolongation structures of nonlinear equations and infinite‐dimensional algebras

Minoru Omote

J. Math. Phys. 27, 2853 (1986); http://dx.doi.org/10.1063/1.527261 (8 pages) | Cited 14 times

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Prolongation structures of the sine–Gordon equation, the Ernst equation, and the chiral model are systematically discussed. It is shown that the prolongation structures generate the Kac–Moody algebra for the sine–Gordon equation and another type of infinite‐dimensional algebra for the Ernst equation. This algebra includes the Kac–Moody algebra and the Virasoro algebra as its subalgebra.
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02.30.Jr Partial differential equations
45.05.+x General theory of classical mechanics of discrete systems
02.20.Sv Lie algebras of Lie groups

On a new hierarchy of nonlinear evolution equations containing the Pohlmeyer–Lund–Regge equation

F. Pempinelli and S. Potenza

J. Math. Phys. 27, 2861 (1986); http://dx.doi.org/10.1063/1.527262 (7 pages) | Cited 1 time

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A hierarchy of local nonlinear evolution equations associated with a new spectral problem is derived. It is shown that each equation is Hamiltonian and that their fluxes commute and a local infinite set of conserved densities is given. An interesting reduction is considered. In this case a hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. Also this hierarchy satisfies a canonical geometrical scheme. It contains as a special case the Pohlmeyer–Lund–Regge equation.
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02.30.Jr Partial differential equations
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Some remarks on the nonlinear integral equation in Kirkpatrick’s theory of glass transition

Tetz Yoshimura

J. Math. Phys. 27, 2868 (1986); http://dx.doi.org/10.1063/1.527263 (4 pages)

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The nonlinear singular integral equation for ‘‘self‐energy’’ Σ(k,z) arising in Kirkpatrick’s mode‐coupling theory of glass transition is analyzed without suppressing the k dependence. An equation that is equivalent to Kirkpatrick’s equation and suitable for high density is set up. Applicability of Lika’s generalization of the Newton–Kantorovich successive approximation is discussed. The possibility of solutions that cannot be found by iteration is pointed out.
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02.30.Rz Integral equations
64.70.P- Glass transitions of specific systems
64.70.Q- Theory and modeling of the glass transition
64.10.+h General theory of equations of state and phase equilibria

A class of continuum models with no phase transitions

David Klein

J. Math. Phys. 27, 2872 (1986); http://dx.doi.org/10.1063/1.527264 (4 pages) | Cited 1 time

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For a restricted family of classical grand canonical continuum interactions, it is proved that the Gibbs state is unique at all temperatures and fugacities. The interactions considered are not translation invariant except in the one‐dimensional case.
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05.70.Fh Phase transitions: general studies
02.50.Cw Probability theory
05.20.Dd Kinetic theory

Central‐limit theorems on groups

P. A. Mello

J. Math. Phys. 27, 2876 (1986); http://dx.doi.org/10.1063/1.527265 (16 pages) | Cited 33 times

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The probability density pN of the product of N statistically independent (and identically distributed, each with probability density p1) elements of a group is studied in the limit N→∞. It is shown, for the compact groups R(2) and R(3), that pN→1 as N→∞, independently of p1. It is made plausible that a similar behavior is to be expected for other compact groups. For noncompact groups, the case of SU(1,1)which is of interest to the physics of disordered conductors, is studied. The case in which p1 is isotropic, i.e., independent of the phases, is analyzed in detail. When p1 is fixed and N≫1, a Gaussian distribution in the appropriate variable is found. When the original variables are rescaled by 1/N and the limit N→∞ is taken, keeping the ratio of the length of the conductor to the localization length fixed, an explicit integral representation for the resulting probability density is found. It is also exhibited that the latter satisfies a ‘‘diffusion’’ equation on the group manifold.
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02.50.Cw Probability theory
02.20.Hj Classical groups
72.15.Cz Electrical and thermal conduction in amorphous and liquid metals and alloys
72.10.Bg General formulation of transport theory

On the DLR equation for the two‐dimensional sine–Gordon model

R. Gielerak

J. Math. Phys. 27, 2892 (1986); http://dx.doi.org/10.1063/1.527266 (11 pages) | Cited 9 times

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The Dobrushin–Lanford–Ruelle equation is studied in a certain space of measures in the case of two‐dimensional trigonometric interactions. The uniqueness theorem extending the results of Albeverio and Hoegh‐Krohn [S. Albeverio and R. Hoegh‐Krohn, Commun. Math. Phys. 68, 95 (1979)] is proved. The extension is obtained by the application of some correlation inequalities of the Ginibre‐type, which reduce the proof of the uniqueness of the translationally invariant, regular, tempered Gibbs states to the question on the independence of the infinite‐volume free energy of the boundary conditions. The required independence is proved in this paper.
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02.50.Ga Markov processes
05.30.Ch Quantum ensemble theory
11.10.-z Field theory

Approximate solution of Fredholm integral equations by the maximum‐entropy method

Lawrence R. Mead

J. Math. Phys. 27, 2903 (1986); http://dx.doi.org/10.1063/1.527267 (5 pages) | Cited 23 times

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An approximate means of solving Fredholm integral equations by the maximum‐entropy method is developed. The Fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum‐entropy methods has been successfully implemented in a previous paper by Mead and Papanicolaou [L. R. Mead and N. Papanicolaou, J. Math. Phys. 25, 2404 (1984)]. Several explicit examples are given of approximate maximum‐entropy solutions of Fredholm integral equations of the first and second kinds and of the Wiener–Hopf type. Both the weaknesses and strengths of the method are discussed.
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02.60.Cb Numerical simulation; solution of equations
02.30.Rz Integral equations

Classical particles with internal structure: General formalism and application to first‐order internal spaces

M. V. Atre and N. Mukunda

J. Math. Phys. 27, 2908 (1986); http://dx.doi.org/10.1063/1.527268 (12 pages) | Cited 9 times

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Group theoretic methods are used to systematically classify all possible internal structures for an elementary classical relativistic particle in terms of coset spaces of SL(2,C) with respect to its continuous subgroups. The allowed internal spaces Q are separated into first‐ and second‐order ones, depending on whether a canonical description can be given using Q itself or if it needs the cotangent bundle T∗Q. Three of the former are found, one corresponding to the use of a Majorana spinor as the internal variable, the other two related to orbits in the Lie algebra of SO(3,1) under the adjoint action. For the latter two, a Lagrangian description of an elementary object with the corresponding internal space is set up, and the dynamics studied.
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45.05.+x General theory of classical mechanics of discrete systems
02.20.Rt Discrete subgroups of Lie groups
11.90.+t Other topics in general theory of fields and particles (restricted to new topics in section 11)
02.20.Sv Lie algebras of Lie groups

Kepler problem with a magnetic monopole

Bruno Cordani

J. Math. Phys. 27, 2920 (1986); http://dx.doi.org/10.1063/1.527269 (2 pages) | Cited 3 times

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It is shown that the usual moment map J: T∗(R3−{0})]so∗(2,4) of the Kepler problem can be generalized to include the magnetic term of the Dirac monopole.
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45.05.+x General theory of classical mechanics of discrete systems
14.80.Hv Magnetic monopoles
11.10.-z Field theory

Quantum kinematics of the harmonic oscillator

Jorge Krause

J. Math. Phys. 27, 2922 (1986); http://dx.doi.org/10.1063/1.527270 (14 pages) | Cited 16 times

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The formalism of non‐Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space‐time realization of the group, casts the Schrödinger operator into the familiar time‐dependent space‐time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time‐dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space‐time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.
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03.65.-w Quantum mechanics
03.65.Ca Formalism
03.65.Fd Algebraic methods
02.20.Sv Lie algebras of Lie groups

Geometric quantization: Modular reduction theory and coherent states

S. Twareque Ali and Gérard G. Emch

J. Math. Phys. 27, 2936 (1986); http://dx.doi.org/10.1063/1.527271 (8 pages) | Cited 15 times

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The natural role played by coherent states in the geometric quantization program is brought out by studying the mathematical equivalence between two physical interpretations that have recently been proposed for this program. These interpretations are based, respectively, on the modular algebra structure of prequantization, and the reproducing kernel structure of phase space quantization. The arguments are presented in this paper for the particular case where the phase space of the system considered is the cotangent bundle T∗M of a homogeneous manifold M, and for didactic reasons, the latter is taken to be a real vector space.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Fd Algebraic methods
45.05.+x General theory of classical mechanics of discrete systems
02.20.Qs General properties, structure, and representation of Lie groups

Coulomb Green’s functions, in an n‐dimensional Euclidean space

L. Chetouani and T. F. Hammann

J. Math. Phys. 27, 2944 (1986); http://dx.doi.org/10.1063/1.527272 (5 pages) | Cited 24 times

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The H‐atom Green’s function is calculated in an n‐dimensional Euclidean space, following the Feynman Lagrangian formulation. The use of generalized polar coordinates allows the expansion of the propagator into partial propagators, and the separation of the angular and radial variables. The angular part is shown to be a generalized Legendre polynomial while the radial part may be transformed in that of the harmonic oscillator. The H‐atom spectrum is given by the poles of the Green’s function.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Ca Formalism
03.65.Db Functional analytical methods

Time‐dependent invariant associated to nonlinear Schrödinger–Langevin equations

Antônio B. Nassar

J. Math. Phys. 27, 2949 (1986); http://dx.doi.org/10.1063/1.527273 (4 pages) | Cited 16 times

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Via the quantum‐hydrodynamical method, a time‐dependent invariant associated to the quantum dissipative time‐dependent harmonic oscillator (TDHO), described by two classes of nonlinear Schrödinger–Langevin equations with the following frictional nonlinear terms is constructed: (i) W1=−iℏν(ln ψ−〈ln ψ〉), which is the Schuch–Chung–Hartman frictional nonlinear term, and (ii) W2=ν{[x−〈x〉][cmath+(1−c)〈 math〉]− 1/2 iℏc}, which includes the Süssmann (c=1), the Hasse (c= 1/2 ), and the Albrecht–Kostin (c=0) frictional nonlinear operators. The associated invariant found is exact for the Schuch–Chung–Hartman and Hasse models, and only approximate for the Süssman and Albrecht–Kostin models.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Db Functional analytical methods
03.65.Sq Semiclassical theories and applications

Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems

C. Batlle, J. Gomis, J. M. Pons, and N. Roman‐Roy

J. Math. Phys. 27, 2953 (1986); http://dx.doi.org/10.1063/1.527274 (10 pages) | Cited 51 times

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The equivalence between the Lagrangian and Hamiltonian formalism is studied for constraint systems. A procedure to construct the Lagrangian constraints from the Hamiltonian constraints is given. Those Hamiltonian constraints that are first class with respect to the Hamiltonian constraints produce Lagrangian constraints that are FL‐projectable.
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03.65.Ca Formalism
45.05.+x General theory of classical mechanics of discrete systems
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Db Functional analytical methods

Chebyshev polynomials and quadratic path integrals

Patrick L. Nash

J. Math. Phys. 27, 2963 (1986); http://dx.doi.org/10.1063/1.527275 (1 page)

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A simple method for the evaluation of path integrals associated with quadratic Lagrangians is discussed. This approach makes use of a relationship between the Van Vleck–Morette determinant and a limit that involves the Chebyshev polynomials of the second kind.
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03.65.Db Functional analytical methods
02.10.De Algebraic structures and number theory

Parametrization of the linear zeros of 6j coefficients

J. J. Labarthe

J. Math. Phys. 27, 2964 (1986); http://dx.doi.org/10.1063/1.527276 (2 pages) | Cited 7 times

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The linear zeros of 6j coefficients are fully parametrized apart from a multiplicative factor in terms of four integers.
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03.65.Fd Algebraic methods
02.10.De Algebraic structures and number theory

Recurrence relations for two‐center harmonic oscillator integrals

J. Morales, L. Sandoval, and A. Palma

J. Math. Phys. 27, 2966 (1986); http://dx.doi.org/10.1063/1.527277 (7 pages) | Cited 10 times

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General recurrence relations for the calculation of two‐center harmonic oscillator (HO) integrals are obtained by means of a hypervirial‐like‐theorem commutator algebra procedure, combined with a second quantization formalism. The method is based on a linear transformation between the creation and annihilation operators of two displaced HO with different frequencies. Ansbacher’s recurrence relations for the calculation of Franck–Condon factors are obtained straightforwardly from the proposed general recurrence relations. The application to polynomial, exponential, and Gaussian operator integrals is shown and new recurrence relations are given. In all cases, the proposed recurrence relations reduce, as particular cases, to the corresponding formulas for the calculation of one‐center integrals.
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03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods
02.30.Rz Integral equations

3+1 Regge calculus with conserved momentum and Hamiltonian constraints

John L. Friedman and Ian Jack

J. Math. Phys. 27, 2973 (1986); http://dx.doi.org/10.1063/1.527224 (14 pages) | Cited 39 times

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The Einstein action is evaluated for space‐times whose three‐metrics on a family of spacelike hypersurfaces are piecewise flat. The 3+1 action of Lund and Regge [F. Lund and T. Regge (private communication)], recently generalized by Piran and Williams [T. Piran and R. M. Williams, Phys. Rev. D 33, 1622 (1986)], is recovered in this way. A natural interpretation of the momentum constraint is obtained for simplicial initial data sets; and, by incorporating a nonzero shift vector and a nonconstant lapse, one finds a formalism in which the constraints are preserved by the time evolution. (In contrast to the continuum case, the constraints are not conserved if the lapse and shift are chosen a priori.) A consistent Hamiltonian formalism is readily obtained by the standard (Bergmann–Dirac) procedure or, alternatively, by algebraically solving the constraint equations for the lapse and shift on each three‐simplex. Explicit solutions to the classical equations are found for spaces built from congruent simplices. In this special case, the action is that of a free relativistic particle moving in a curved space‐time with indefinite metric and a conformal timelike Killing vector. For general space‐times, if one a priori sets the shift vector to zero, the action has the form of a sum of such free‐particle actions, but one for which the different particles interact by having coordinates in common.
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04.20.Cv Fundamental problems and general formalism
04.20.-q Classical general relativity
11.55.Jy Regge formalism

Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids

R. Maartens, D. P. Mason, and M. Tsamparlis

J. Math. Phys. 27, 2987 (1986); http://dx.doi.org/10.1063/1.527225 (8 pages) | Cited 40 times

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An example from a perfect fluid FRW space‐time is presented to show that a conformal Killing vector (CKV) need not map fluid flow lines into fluid flow lines. Kinematic properties of the Lie derivative along a CKV of timelike and spacelike unit vectors are derived and applied to the fluid unit four‐velocity vector. Dynamic properties of special conformal Killing vectors (SCKV) in a fluid with anisotropic pressure and vanishing energy flux are obtained using Einstein’s field equations. It is shown that a SCKV maps both fluid flow lines and integral curves of na into themselves, where na is the unit spacelike vector of anisotropy. The relation between the anisotropic pressure components and the energy density is considered. By means of an example from a radiationlike viscous fluid FRW space‐time it is shown that the dynamic results depend crucially on the vanishing of the energy flux vector. The extension of the dynamic results to a fluid with arbitrary stress tensor and zero energy flux vector is examined.
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04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime
04.20.Jb Exact solutions

Power law singularities in the scale covariant theory

A. Beesham

J. Math. Phys. 27, 2995 (1986); http://dx.doi.org/10.1063/1.527226 (3 pages) | Cited 1 time

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Power asymptote singularities are discussed in the scale covariant theory of gravitation. Some general results are derived. Special attention is paid to the Friedmann [A. Friedmann, Z. Phys. 10, 377 (1922)] and Kasner [E. Kasner, Am. J. Math. 43, 217 (1921)] models. A wider class of behavior is exhibited and it is shown that the results obtained constitute a generalization of the corresponding general relativistic results.
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04.50.-h Higher-dimensional gravity and other theories of gravity
04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

The unboundedness of the gravitational partition function

B. Broda

J. Math. Phys. 27, 2998 (1986); http://dx.doi.org/10.1063/1.527227 (5 pages)

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The unboundedness of the gravitational partition function Z is formally established. First the Euclidean path integral representation for Z, in terms of the renormalized effective action Seffren, is derived. Next it is shown that for ‘‘strong’’ fields, Seffren is unbounded from below. The possible influence of the space‐time topology is taken into account.
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04.60.-m Quantum gravity
02.40.Ky Riemannian geometries
02.40.Pc General topology
04.20.Cv Fundamental problems and general formalism
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