Volume 28, Issue 11, November 1987
Index of content:

Totally symmetric irreducible representations of the group SO(6) in the principal SO(3) subgroup basis
View Description Hide DescriptionExplicit matrix elements are found for the generators of the group SO(6) in an arbitrary totally symmetric irreducible representation, using the physical principal SO(3) subgroup in the chain SO(6)⊇SO(5)⊇SO(3). The internal one missing label problem is solved through the definition of intrinsic states associated to the SU(2)×SU(2) subgroup in the chain SO(6)⊇SO(5)⊇SU(2)×SU(2) and out of which is projected a complete set of states in the physical basis by integrations over the physical rotation group manifold. The matrix elements of the SO(6) generators in the SU(2)×SU(2) basis are themselves obtained by the intermediate use of an SU(2)×SU(2)×U(1) basis, the latter group being a subgroup of SO(6) but not of SO(5).

Theorems on the Jordan–Schwinger representations of Lie algebras
View Description Hide DescriptionThe Jordan–Schwinger representations of Lie algebras are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. When the representation is Hermitian, the Lie algebra is shown to be unitary for the fermion case and pseudounitary for the boson case. It is also shown that the representation of a Lie algebra leads to a projective representation of the Lie group corresponding to the Lie algebra.

Canonical map approach to channeling stability in crystals. II
View Description Hide DescriptionA nonrelativistic and a relativistic classical Hamiltonian model of two degrees of freedom are considered describing the plane motion of a particle in a potential V(x _{1},x _{2})[(x _{1},x _{2}) =Cartesian coordinates]. Suppose V(x _{1},x _{2}) is real analytic in its arguments in a neighborhood of the line x _{2}=0, one‐periodic in x _{1} there, and such that the average value of ∂V(x _{1},0)/∂x _{2} vanishes. It is proved that, under these conditions and provided that the particle energy E is sufficiently large, there exist for all time two distinguished solutions, one satisfying the equations of motion of the nonrelativistic model and the other those of the relativistic model, whose corresponding configuration‐space orbits are one‐periodic in x _{1} and approach the line x _{2}=0 as E→∞. The main theorem is that these solutions are (future) orbitally stable at large enough E if V satisfies the above conditions, as well as natural requirements of linear and nonlinear stability. To prove their existence, one uses a well‐known theorem, for which a new and simpler proof is provided, and properties of certain natural canonical maps appropriate to these respective models. It is shown that such solutions are orbitally stable by reducing the maps in question to Birkhoff canonical form and then applying a version of the Moser twist theorem. The approach used here greatly lightens the labor of deriving key estimates for the above maps, these estimates being needed to effect this reduction. The present stability theorem is physically interesting because it is the first rigorous statement on the orbital stability of certain channeling motions of fast charged particles in rigid two‐dimensional lattices, within the context of models of the stated degree of generality.

A formula on the Wiener–Hermite expansion
View Description Hide DescriptionTwo mathematical formulas are given in an explicit form: one develops a product of two different multiple Wiener integrals in a series of Wiener–Hermite expansions; the other one develops a product of two random variables, each of which is described by a Wiener–Hermite expansions, in a series of another Wiener–Hermite expansion.

Spontaneous splitting and internal isometries of superstring vacua
View Description Hide DescriptionSuperstring vacua are normally presumed to be of the form M×K, where dim(M)=4, dim(K)=6, and where × denotes the g l o b a l Riemannian product. Since, however, one would ultimately wish to understand the external/internal distinction in terms of some dynamical mechanism (‘‘spontaneous splitting’’) involving vacuum expectation values of local fields, it may be preferable to use a l o c a l Riemannian product at the outset. Here it is shown that these spaces, which have the same local (block‐diagonal) type of metric as M×K, can be described and classified by examining the isometry groups of the Calabi–Yau manifolds which have been proposed as models for the internal superstring vacua.

Canonical stochastic dynamical systems
View Description Hide DescriptionA canonical stochastic dynamical system for time‐symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.

Deformation of algebras and solution of self‐duality equation
View Description Hide DescriptionThe solutions of the self‐duality equation depending on a set of functions of three independent variables are constructed in an explicit way. The obtained solutions are shown to be connected with two‐dimensional systems determined by the operators of the Lax pair.

On a new class of completely integrable nonlinear wave equations. II. Multi‐Hamiltonian structure
View Description Hide DescriptionThe multi‐Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi‐Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi‐Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri‐Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson’s equation in nonlinear acoustics.

A search for bilinear equations passing Hirota’s three‐soliton condition. III. Sine–Gordon‐type bilinear equations
View Description Hide DescriptionIn this paper the results of a search for pairs of bilinear equations of the type A ^{ i }(D _{ x },D _{ t })F⋅F +B ^{ i }(D _{ x },D _{ t })G⋅F +C ^{ i }(D _{ x },D _{ t })G⋅G=0, i=1,2, which have standard type three‐soliton solutions, are presented. The freedom to rotate in (F,G) space is fixed by the one‐soliton ansatz F=1, G=e ^{ n }, then the B ^{ i } determine the dispersion manifold while A ^{ i } and C ^{ i } are auxiliary functions. In this paper it is assumed that B ^{1} and B ^{2} are even and proportional, and that A ^{ i } and C ^{ i } are quadratic. As new results, B ^{1}=a D ^{3} _{ x } D _{ t } +D _{ t } D _{ y }+b, A ^{2}=−C ^{2}=D _{ x } D _{ t }, and generalizations of the sine–Gordon model B ^{1}=D _{ x } D _{ t }+a with a family of auxiliary functions A ^{ i } and C ^{ i } are obtained.

Nonlinear resonance for quasilinear hyperbolic equation
View Description Hide DescriptionThe purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source. Resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system. For the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena. Motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source. Analytical understanding is used to design a numerical scheme based on the random choice method. An important physical example is transonic gas flow through a nozzle. This analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow, including such phenomena as nonlinear stability, instability, and changing types of waves.

Reflection of waves in nonlinear integrable systems
View Description Hide DescriptionSolutions of the boomeron type are found in two nonlinear integrable systems describing the interaction of a long wave with a short wave packet. These solutions follow from two‐soliton solutions if certain additional conditions are imposed on their parameters. The results are relevant to some problems of plasma physics, solid‐state physics, hydrodynamics, etc.

Yang–Mills solutions in S ^{3}×S ^{1}
View Description Hide DescriptionA search is conducted for new solutions to Yang–Mills classical field theory defined on S ^{3}×S ^{1} using Witten’s ansatz. Then some sort of non‐Abelian plane waves giving rise to a divergent energy are obtained.

On total noncommutativity in quantum mechanics
View Description Hide DescriptionIt is shown within the Hilbert space formulation of quantum mechanics that the total noncommutativity of any two physical quantities is necessary for their satisfying the uncertainty relation or for their being complementary. The importance of these results is illustrated with the canonically conjugate position and momentum of a free particle and of a particle closed in a box.

The connection of two‐particle relativistic quantum mechanics with the Bethe–Salpeter equation
View Description Hide DescriptionThe formal equivalence between the wave equations of two‐particle relativistic quantum mechanics, based on the manifestly covariant Hamiltonian formalism with constraints, and the Bethe–Salpeter equation are shown. This is achieved by algebraically transforming the latter so as to separate it into two independent equations that match the equations of Hamiltonian relativistic quantum mechanics. The first equation determines the relative time evolution of the system, while the second one yields a three‐dimensional eigenvalueequation. A connection is thus established between the Bethe–Salpeter wave function and its kernel on the one hand and the quantum mechanical wave function and interaction potential on the other. For the sector of solutions of the Bethe–Salpeter equation having nonrelativistic limits, this relationship can be evaluated in perturbation theory. A generalized form of the instantaneous approximation that simplifies the various expressions involved in the above relations is also devised. It also permits the evaluation of the normalization condition of the quantum mechanical wave function as a three‐dimensional integral.

Boson algebra as a symplectic Clifford algebra
View Description Hide DescriptionIt is well known that the n‐fermion algebra is a complex Clifford algebra of dimension 2^{2n } with the orthogonal group O(2n,C) as group of automorphisms. The n‐boson algebra viewed similarly as a complex ‘‘symplectic Clifford’’ algebra is investigated. It is of infinite dimension and has the symplectic group Sp(2n,C) as a group of automorphisms. Special attention is given to the case n=1 and various bases. In particular, the matrix elements of the bases between harmonic oscillator states and their relation with special functions are investigated.

Factorizations of vector operators for the isotropic harmonic oscillator in an angular momentum basis
View Description Hide DescriptionThe factorization of four vector operators, D ^{±}(ω) and D ^{±}(−ω), which occur in a representation‐independent, spectrum‐generating algebra for the three‐dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D ^{±}(ω) are quantum‐mechanical analogs of the classical vectors (1∓i L̂×)F _{ c } (ω), where F _{ c }(ω)=−Mωr×L+p L is constant in a frame rotating with angular velocity ωL̂. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L ^{2}), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖l m〉 of L ^{2} and L _{ z }. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) The remaining eight operators factorize linear combinations of the Hamiltonian and the dimension operator. In linearized, differential form four of these operators are ladder operators for energy and angular momentum in the radial part of the coordinate‐space wave functions, while the other four are the corresponding operators in the momentum representation.

Probability of convergence of perturbation theory for hydrogen photoionization
View Description Hide DescriptionThe probability is estimated that at a given frequency ω the action of an external, monochromatic radiation field acting on a hydrogen atom yields a fixed positive convergence radius of the Rayleigh‐Schrödinger perturbation expansion for the quasienergy resonances.

Canonical transformations and exact invariants for dissipative systems
View Description Hide DescriptionA simple treatment to the problem of finding exact invariants and related auxiliary equations for time‐dependent oscillators with friction is presented. The treatment is based on the use of a time‐dependent canonical transformation and an auxiliary transformation.

Evaporation of nonzero rest mass particles from a black hole
View Description Hide DescriptionAnalytic expressions for the transmission coefficient and the emission and the absorption rates for scalar particles with mass and a chargeless, nonrotating black hole are calculated by using Jacobian elliptic functions and integrals in the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation.

Nonorientable one‐loop amplitudes for the bosonic open string: Electrostatics on a Möbius strip
View Description Hide DescriptionThe partition function, N‐point scalar, and four‐point vector nonorientable one‐loop amplitudes for the bosonic open string in the critical dimension are obtained using a first quantized path integral treatment of Polyakov’s string that assumes scale independence.