Volume 28, Issue 8, August 1987
Index of content:

A folk theorem revisited: Degenerate representations
View Description Hide DescriptionIt is known that when bases for generic irreducible representations of a semisimple group are reduced according to a semisimple subgroup the number of functionally independent missing label operators is just twice the number of missing labels. It is shown that the relation continues to hold when degenerate irreducible representations are considered.

Indices of representations of simple superalgebras
View Description Hide DescriptionIndices and anomaly numbers for representations of basic classical Lie superalgebras are defined, and their explicit expressions are derived in terms of Kac–Dynkin labels. Useful properties of indices and anomalies are determined, and several examples are given. A similar analysis is made for superindices and superanomalies, and it is demonstrated how all these objects form a helpful tool in decomposing tensor products or in constructing branching rules for representations.

Parent potentials for an infinite class of reflectionless kinks
View Description Hide DescriptionThe sine–Gordon and φ‐four kinks are known to be reflectionless by virtue of the fact that their small oscillations are governed by the modified Pöschl–Teller potential U _{ l }(x) =1−[(l+1)/l]sech^{2}(x/l), with l=1 and 2, respectively. An infinite class of parent potentials V _{ l }(φ) analogous to V _{1}∼1−cos φ for sine–Gordon kinks and V _{2}∼(1−φ^{2})^{2} for φ‐four kinks, which bear reflectionless kinks, are constructed. This is done by requiring the lowest bound‐state eigenfunction of U _{ l }(x) to be proportional to the spatial derivative of the kink waveform φ^{(l)} _{ K }(x), i.e., the translational mode of the kink. The resulting differential equation is solved for V _{ l }(φ) to find that it can be expressed in terms of the student’s t distribution of probability theory. Various properties of the parent potentials and their reflectionless kinks are discussed.

On the τ‐functions of A ^{(2)} _{2}
View Description Hide DescriptionA general construction of partial differential equations satisfied by the components of τ‐functions is given by considering the tensor products of modules. This procedure is applied to the A ^{(2)} _{2}‐ modules L(Λ_{0}) and L(Λ_{1}), leading to the Kaup equation and Sawada–Kotera equation, respectively. Although L(Λ_{0}) and L(Λ_{1}) are of different level, one can consider L(Λ_{0})⊗L(Λ_{1}), leading to so‐called modified equations. This last construction is new, and leads to a different choice of y, the variable that generates the equations.

On the prolongation approach in three dimensions for the conservation laws and Lax pair of the Benjamin–Ono equation
View Description Hide DescriptionThe prolongation structure approach of Wahlquist and Estabrook [J. Math. Phys. 1 6, 1 (1975)] is used effectively in a new situation in relation to the integrodifferential type BO equation (the Benjamin–Ono equation). The main clue lies in the possible differential equation representation of such equations in three dimensions. Here it is shown how the usual analysis of prolongation structure can be utilized to deduce a Lax pair for a BO type equation in three dimensions. Effectiveness of the present approach is further demonstrated by an independent derivation of some conservation laws associated with the equation. Last, the whole formalism is reduced to two dimensions to make contact with known results.

Generalized complex superspace—Involutions of superfields
View Description Hide DescriptionDistinguished involutions called transposition and conjugation in the algebra of complex supernumbers are introduced. On the base of these involutions an analysis is developed over complex superspaces. Consistency of integration under supertransformations between two complex superspaces is shown.

Cohomology in connection space, family index theorem, and Abelian gauge structure
View Description Hide DescriptionUsing the natural connection form on a principal bundle P(M,G) and the bundle A(A/G,G) a systematic derivation of the double‐cohomological series constituted by the exterior differential d on space‐time M and arbitrary, horizontal, and vertical variations in connection space is given. The relationship between these cohomologies and the family index theorem is clarified. The formalism is then used to analyze Abelian gauge structure inside non‐Abelian gauge theory. The pertinent functionalU(1) connection form, curvature form, and three‐form ‘‘curvature’’ are identified and computed, and are related to the θ vacuum, anomalous commutation relation, and Jacobi identity, respectively. Some of the results differ from those obtained by Wu and Zee [Nucl. Phys. B 2 5 8, 157 (1985)] and Niemi and Semenoff [Phys. Rev. Lett. 5 5, 227 (1985)] and the results of this paperrecover theirs under certain conditions. Finally the generalization of the formalism to a nontrivial principal bundle by introduction of a fixed background connection form is discussed.

Large rectangular random matrices
View Description Hide DescriptionModels with a multiplet of field variables arranged into rectangular matrices, in the limit of infinite dimensions of the matrices, are studied. In zero‐dimensional space (where the problem is a combinatorial one) a closed solution is given that improves the one previously known. In arbitrary space dimension a symmetry is described that connects rectangular models with vector models.

Waves, bifurcations, and solitons in a model with sixfold symmetry
View Description Hide DescriptionThe traveling‐wave solutions of a model Lagrangian density for a complex‐valued scalar field with possible application to charge density waves in systems with sixfold symmetry are explored by means of the slow‐fluctuation technique. There are five main wave types separated by three bifurcations with respect to integration constants. In addition, there are several classes of degenerate waves, including some with purely harmonic, stable, straight‐line or circular motions of the amplitude vector. Beyond the range of the slow‐fluctuation approximation there are several unstable soliton solutions, three of which have unexpectedly curved paths of the amplitude vector.

On head‐wave amplitudes
View Description Hide DescriptionThe head‐wave contribution to a reflection is investigated by two different methods and it is shown that the new result presented by Lerche and Hill [J. Math. Phys. 2 6, 1420 (1985)] for the head‐wave amplitude is in error due to the use of an inappropriate mathematical method.

A search for bilinear equations passing Hirota’s three‐soliton condition. I. KdV‐type bilinear equations
View Description Hide DescriptionIn this paper the results of a search for bilinear equations of the type P(D _{ x }, D _{ t })F ⋅ F=0, which have three‐soliton solutions, are presented. Polynomials up to order 8 have been studied. In addition to the previously known cases of KP, BKP, and DKP equations and their reductions, a new polynomialP=D _{ x } D _{ t }(D _{ x } ^{2} +√3D _{ x } D _{ t }+D _{ t } ^{2}) +a D _{ x } ^{2}+b D _{ x } D _{ t } +c D _{ t } ^{2} has been found. Its complete integrability is not known, but it has three‐soliton solutions. Infinite sequences of models with linear dispersion manifolds have also been found, e.g., P=D _{ x } ^{ M } D _{ t } ^{ N } D _{ y } ^{ P }, if some powers are odd, and P=D _{ x } ^{ M } D _{ t } ^{ N }(D _{ x } ^{2} −1)^{ P }, if M and N are odd.

Spinor focus wave modes
View Description Hide DescriptionNew solutions of the homogeneous spinor wave equation are obtained. They are similar to the focus wave mode solutions of Maxwell’sequations leading to a Gaussian pulse energy. A weighted superposition of these modes may supply finite energy pulses. The particular case of Bessel weight functions is discussed.

The anharmonic oscillator at a finite temperature. Comparison of quantum and classical stochastic calculations
View Description Hide DescriptionAn oscillator with a small, but otherwise arbitrary, perturbing potential is considered immersed in a random cavity radiation. Classical (stochastic) calculations are done when the radiation has a Rayleigh–Jeans spectrum and a complete Planck spectrum (i.e., with zero point). These are compared with the results obtained by a quantum calculation. First, a comparison is made of stationary values, in particular, the energy. Then the emission and the absorption spectra are calculated, in particular, the absorptionspectrum for an arbitrary incoming radiation. Finally, a detailed comparison is made of the absorption bands when the perturbing potential has the form λx ^{2K } (K=2,3,...). In all cases, it is explicitly shown that the quantum and the classical behavior agree in the limit of high temperatures. It is also shown that the classical system immersed in a radiation with complete Planck spectrum is much closer to the quantum system than the fully classical system (with a Rayleigh–Jeans spectrum).

Direct calculation of the Berry phase for spins and helicities
View Description Hide DescriptionThe Berry phase for spins or helicities is calculated in a simple way in which it appears more as a property of the spin states than of the Hamiltonian. The calculation applies to nonrelativistic particles or relativistic particles with either zero or nonzero mass. A simple way to see how the Berry phase corresponds to rotation of the electric and magnetic fields of plane‐polarized light is also pointed out.

Compatibility of observables represented by positive operator‐valued measures
View Description Hide DescriptionThe proof of a result analogous to that in Koelman and de Muynck [Phys. Lett. A 9 8, 1 (1983)] is given for the case of unbounded observables. If two, not necessarily bounded, observables are represented by a positive operator‐valued measure, then the measurement of any of them is undisturbed if and only if they commute. The Naimark theorem on dilations of spectral functions is exploited. A stronger version of Wigner’s theorem is given.

Heisenberg inequality and the complex field in quantum mechanics
View Description Hide DescriptionObservables A and B satisfy the Heisenberg inequality if the product of their variances has a positive lower bound independent of the state of the system. In the Hilbert space formulation of quantum mechanics it is a consequence of the Schwarz inequality that the Heisenberg‐type inequality Var(A,φ)⋅Var(B,φ)≥ (1)/(4) ‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖^{2} holds for any pair of observables A and B (represented as self‐adjoint operators) and for any (vector) state (represented as a unit vector). If inf{‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖ ‖φ∈dom(A) ∩dom(B)}≠0 then A and B satisfy the Heisenberg inequality. In the present paper the derivability of the Heisenberg‐type inequality is analyzed within the general theoretical frame of a sum logic. It is shown that any real‐valued non‐negative function (A,α)→f(A,α) of observables A and of states α, which has a symmetry property f(A+B,α)+f(A−B,α)=2f(A,α)+2f(B,α) with respect to observables, satisfies the Heisenberg‐type inequalityf(A,α)⋅f(B,α)≥ (1)/(4) ‖ f(A+B,α)−f(A,α)−f(B,α)‖^{2} for all observables A and B and for all states α. The natural probabilistic realizations f _{1}(A,α)=Exp(A ^{2},α) and f _{2}(A,α)=Var(A,α) of such functions are then analyzed. It turns out that only with the complex extension of the theory can the Heisenberg inequality be attained in the Hilbert space realization of the theory. This is used as an argument in favor of the complex field as the scalar field of quantum mechanics.

Maximal and minimal eigenvalues and their associated nonlinear equations
View Description Hide DescriptionThe spectral theory of uniformly elliptic operators A under perturbations V giving rise to operators of the form H _{ V }=A+V(x) on a bounded or unbounded region, such as Schrödinger operators, are considered. Suppose that ∥V∥_{ p } is constrained, but V is otherwise unspecified. The theory of the potentials V that maximize or minimize the eigenvalues of H _{ V } is presented. The optimizing potentials are typically determined by equations of the form −Δu+W(x)u=±c u ^{α}+Λu. The optimization of eigenvalues also turns out to be related to the determination of the best constants in Sobolev’s inequality, and, in its one‐dimensional simplification, to a classical oscillator problem with ‘‘instanton’’ properties.

An application of filtering theory to parameter identification using stochastic mechanics
View Description Hide DescriptionAn estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple illustrations, this method is used to obtain estimates for the initial momentum of a free particle given measurements on its trajectory and to construct an estimator for the unknown parameters in a harmonic potential. It is shown that an optimal estimator exists if the stochastic processes are associated with a wave function from a potential of the Rellich type. In addition the a p o s t e r i o r iprobability density of the parameters in the quantal system is calculated, assuming that all parameters involved prescribe a Rellich potential.

An application of filtering theory to parameter identification in quantum mechanics
View Description Hide DescriptionA method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a p o s t e r i o r iprobability density if the original quantal system does not contain any random elements. This probability density for the unknown matrix elements is calculated, and it is demonstrated that for a full Hamiltonian matrix the asymptotic variance of the parameter estimator decreases as a third power in time and a fourth power in the initial conditions. Some differences with the minimum least‐square method are mentioned, and a few issues of numerical implementation are discussed.

Perturbative results from the 1/N expansion for screened Coulomb potentials
View Description Hide DescriptionThe energy eigenvalues for the Hulthén, Yukawa, and exponential‐cosine screened Coulomb potentials are calculated in the 1/N expansion. States with up to three nodes in the wave functions are considered. We obtain the perturbative results up to the order of λ^{8}, where λ is the screening parameter.