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Volume 27, Issue 12, December 1986

Irreducible *‐representations of Lie superalgebras B(0,n) with finite‐degenerated vacuum
View Description Hide DescriptionThe problem of getting irreducible *‐representations π of Lie superalgebrasB(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 C^{ N } ‐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 L ^{2}(R^{+})⊗C^{2} 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.

The generalized atypical supertableaux of the orthosymplectic groups OSP(2‖2p)
View Description Hide DescriptionThe 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.

Principal five‐dimensional subalgebras of Lie superalgebras
View Description Hide DescriptionThe analog of sl(2) for Lie superalgebras is osp(1,2), a five‐dimensional superalgebra. All basic classical Lie superalgebrasL 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.

Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable?
View Description Hide DescriptionThe 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.

Prolongation structures of nonlinear equations and infinite‐dimensional algebras
View Description Hide DescriptionProlongation 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.

On a new hierarchy of nonlinear evolution equations containing the Pohlmeyer–Lund–Regge equation
View Description Hide DescriptionA 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.

Some remarks on the nonlinear integral equation in Kirkpatrick’s theory of glass transition
View Description Hide DescriptionThe 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.

A class of continuum models with no phase transitions
View Description Hide DescriptionFor 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.

Central‐limit theorems on groups
View Description Hide DescriptionThe probability densityp _{ N } of the product of N statistically independent (and identically distributed, each with probability densityp _{1}) elements of a group is studied in the limit N→∞. It is shown, for the compact groups R(2) and R(3), that p _{ N }→1 as N→∞, independently of p _{1}. 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 p _{1} is isotropic, i.e., independent of the phases, is analyzed in detail. When p _{1} 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.

On the DLR equation for the two‐dimensional sine–Gordon model
View Description Hide DescriptionThe 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. 6 8, 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.

Approximate solution of Fredholm integral equations by the maximum‐entropy method
View Description Hide DescriptionAn 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. 2 5, 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.

Classical particles with internal structure: General formalism and application to first‐order internal spaces
View Description Hide DescriptionGroup 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.

Kepler problem with a magnetic monopole
View Description Hide DescriptionIt is shown that the usual moment map J: T*(R^{3}−{0})^{]}so*(2,4) of the Kepler problem can be generalized to include the magnetic term of the Dirac monopole.

Quantum kinematics of the harmonic oscillator
View Description Hide DescriptionThe 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.

Geometric quantization: Modular reduction theory and coherent states
View Description Hide DescriptionThe 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 algebrastructure 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 manifoldM, and for didactic reasons, the latter is taken to be a real vector space.

Coulomb Green’s functions, in an n‐dimensional Euclidean space
View Description Hide DescriptionThe 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.

Time‐dependent invariant associated to nonlinear Schrödinger–Langevin equations
View Description Hide DescriptionVia 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) W _{1}=−iℏν(ln ψ−〈ln ψ〉), which is the Schuch–Chung–Hartman frictional nonlinear term, and (ii) W _{2}=ν{[x−〈x〉][c p̂+(1−c)〈 p̂〉]− 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 e x a c t for the Schuch–Chung–Hartman and Hasse models, and only a p p r o x i m a t e for the Süssman and Albrecht–Kostin models.

Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems
View Description Hide DescriptionThe 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.

Chebyshev polynomials and quadratic path integrals
View Description Hide DescriptionA 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.

Parametrization of the linear zeros of 6j coefficients
View Description Hide DescriptionThe linear zeros of 6j coefficients are fully parametrized apart from a multiplicative factor in terms of four integers.