Index of content:
Volume 21, Issue 3, March 1980

The planar approximation. II
View Description Hide DescriptionThe planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is illustrated on a simple model coupling two N×N matrices.

On the SU_{2} unit tensor
View Description Hide DescriptionThis paper deals with the SU_{2} ⊆ G unit tensor operators t _{ kμα}. In the case where the spinor point group G coincides with U _{1}, then t _{ kμα} reduces (up to a constant) to the (Wigner–Racah–Schwinger) tensor operator t _{ k qα}, an operator which produces an angular momentum state ‖ j+α, m+q〉 when acting upon the state ‖ j m〉. We first investigate those general properties of t _{ kμα} which are independent of their realization. We then turn our attention to realizations of t _{ kμα} in terms of two pairs of boson creation and annihilation operators. This leads us to look at the Schwinger calculus (found to be connected to the de Sitter algebra so_{3,2}) relative to one angular momentum or two coupled angular momenta. As a by‐product, we give a procedure for producing recursion relationships between SU_{2} ⊆ U _{1} Wigner coefficients. Expressions for t _{ k qα}, which cover the cases k integer and half‐an‐odd‐integer, are derived in terms of boson operators. When k is integer, the latter expressions can be rewritten in the enveloping algebra of so_{3} or so_{3,2} according to as α=0 or α≠0. Finally, we study in two appendices some of the properties of (i) the Wigner and Racah operators for an arbitrary compact group and (ii) the SU_{2} ⊆ G coupling coefficients.

Projective covering group versus representation groups
View Description Hide DescriptionThe relation between two methods of finding the unitary projective representations of a connected Lie group is studied. It is shown that the one of the projective covering group G̃ is singled out as the simplest one, but one could also get a connected and simply connected Lie splitting group of minimal dimension by using a similar procedure.

On an infinitesimal approach to semisimple Lie groups and raising and lowering operators of O(n) and U(n)
View Description Hide DescriptionA purely algebraic approach to the evaluation of the fundamental Wigner coefficients and reduced matrix elements of O(n) and U(n) is given. The method employs the explicit use of projection operators which may be constructed using the polynomial identities satisfied by the infinitesimal generators of the group. As an application of this technique, a certain set of raising and lowering operators for O(n) and U(n) are constructed. They are simpler in appearance than those previously constructed since they may be written in a compact product form. They are, moreover, Hermitian conjugates of one another, and therefore are easily normalized.

The KdV prolongation algebra
View Description Hide DescriptionIt is shown that the Wahlquist–Estabrook prolongation of the KdV equation determines a Lie algebraL and that L is infinite‐dimensional. Finite‐dimensional prolongation algebras are shown to be homomorphic images of L. Two such algebras—SL(2,IR) and a five‐dimensional nilpotent algebraH are obtained as quotients of L by ideals which are respectively maximal and maximal for a given property. L determines a connection on a trivial vector bundle and the connection is flat iff the KdV equation is satisfied. The symmetries of the KdV equation are used to construct automorphisms of L and it is shown that H is distinguished by these automorphisms. It is demonstrated that H is closely related to the three classical KdV conservation laws and that the sequential prolongation process for KdV yields only representations of H.

Integration of near‐resonant systems in slow‐fluctuation approximation
View Description Hide DescriptionAutonomous, conservative, nonlinear oscillatory systems of many degrees of freedom with an ’’internal resonance,’’ i.e., with the linearized (’’normal’’) frequencies tied in a near‐resonant relation Jg _{ i }ω_{ i }=−, can be completely integrated in a transparent approximation which is a variant of the ’’method of slowly variable amplitude and phase.’’ It consists in developing the nonlinear coupling into a trigonometric polynomial and dropping all terms except the most slowly varying ones on the grounds that their perturbative effects are oscillatory but essentially not cumulative. In this ’’slow‐fluctuation’’ technique, the amplitudes separate off the phases; they exhibit joint extrema governed by simple conservation laws and all follow from one single quadrature. The phases also require no more than quadratures. There are various possibilities of motion with all amplitudes constant and harmonic time variation at (generally) incommensurate frequencies. The approximation does not apply if a degree of freedom which occurs to the first power in the nonlinear coupling happens to dip down towards zero amplitude; abrupt phase changes can then take place and call for a separate, accurate integration covering the critical time interval. With this sole restriction the method is entirely general. It is exact in the small‐amplitude limit; at finite amplitudes it differs from the exact solution for practical purposes by no more than minor high‐frequency ripples; and it yields more physical insight than any other known method for less effort.

Path integral in the representation of SU(2) coherent state and classical dynamics in a generalized phase space
View Description Hide DescriptionPath integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two‐dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.

Is the Feynman–Dyson series adequate for the aysmptotic expansion of the Green’s function of the quantum mechanical anharmonic oscillator?
View Description Hide DescriptionWe discuss the asymptotic expansion of the Green’s function of the quantum mechanical anharmonic oscillator by applying the method of steepest descents in the evaluation of the corresponding Euclidean path integrals. The contributions arising from the pseudoparticle solutions are then shown to be zero.

A set of commuting operators and R(3) scalars for the complete classification of quadrupole phonon‐states
View Description Hide DescriptionA method is developed for constructing operator forms which commute with the Casimir operators of the groups appearing in the chain U(5) ⊆SU(5) ⊆R(5) ⊆R(3) ⊆ (2), according to which the quadrupole‐phonon states are usually classified. An expression for such operator is given.

Inverse scattering. I. One dimension
View Description Hide DescriptionThis paper presents two new methods of reconstructing an underlying potential in the one‐dimensional Schrödinger equation from a given S matrix. One of these methods is based on a Gel’fand–Levitan equation, the other on a Marchenko equation. A sequel of this paper will treat the three‐dimensional case by similar methods.

Combinatorics, partitions, and many‐body physics
View Description Hide DescriptionSome combinatorial techniques are presented which streamline the graphical analysis used in N‐body scattering theory. The basic results are derived using properties of the lattice of partitions of N particles, which naturally arises on classifying translational symmetry properties of N‐body operators. Classical cumulant expansions are recovered, previously obtained results are presented from a unified point of view, and some new theorems concerning connectivity of N‐body equations are presented.

Field equations in twistors
View Description Hide DescriptionAs part of the twistor program for the ’’quantization of relativity’’ the physical field equations have been translated into twistors.

Inhomogeneous rotating universes with closed timelike geodesics of matter
View Description Hide DescriptionWe present a new class of inhomogeneous stationary cosmological solutions of Einstein–Maxwell equations, with rotating dust and electromagnetic fields. For a subclass of these models, the topology of the space–time manifold is S ^{3}×R, and the timelike geodesic lines of dust are closed; electromagnetic fields are necessary only to avoid matter singularities, while in the completely homogeneous limit they are essential for consistency of field equations; these electromagnetic fields have the structure of magnetic monopole fields, since the total magnetic flux across the unit 2‐sphere is different from zero—that is, magnetic monopoles must be present as a source of the fields.

On the role of space–time topology in quantum phenomena: Superselection of charge and emergence of nontrivial vacua
View Description Hide DescriptionSchwarzschild–Kruskal space–time admits a two‐parameter family of e v e r y w h e r e r e g u l a r, static, source‐free Maxwell fields. It is shown that there exists a corresponding two‐parameter family of unitarily inequivalent representations of the canonical commutation relations. Elements of the underlying Hilbert space may be interpreted as ’’quantum fluctuations of the Maxwell field off nontrivial classical vacua.’’ The representation corresponding to the ’’trivial’’ sector—i.e., the zero classical solution— is the usual Fock representation. All others are ’’non‐Fock.’’ In particular, in all other sectors, the Maxwell field develops a nonzero vacuum expectation value. The parameters labelling the family can be interpreted as electric and magnetic charges. Therefore, unitary inequivalence naturally leads to superselection rules for these charges. These features arise in spite of the linearity of field equations only because the space–time topology is ’’nontrivial.’’ Also, because of linearity, an exact analysis is possible at the quantum level; recourse to perturbation theory is unnecessary.

Analytic properties of the critical dynamics ε expansion
View Description Hide DescriptionThe ε‐expansion technique for evaluating certain integrals in critical dynamics is studied in detail. The analytic properties in the complex ε plane are exploited to get improved answers at the physical value of ε. A different scheme—the ? expansion—is suggested. This has a simpler analytic structure and thus gives more reliable results.

On exact solutions of the nonlinear Heisenberg–Klein–Gordon equation in a space–time of constant spacelike curvature
View Description Hide DescriptionA nonlinear complex s c a l a rfield theory associated with a ’’squared’’ H e i s e n b e r g–P a u l i–W e y l n o n l i n e a r s p i n o r e q u a t i o n is considered. In a d+1 dimensional universe of constant spatial curvature e x a c t localized solutions for the resulting [‖φ‖^{2d/(d−2)}−const ‖φ‖^{2(d−1)/(d−2)}] model are constructed. For ’’soliton‐like’’ solutions with quantized (nontopological) charge the field energy and the Heisenberg uncertainty principle are analyzed.

Finite field equation of Yang–Mills theory
View Description Hide DescriptionWe consider the finite local field equation −{[1+1/α (1+f _{4})]g ^{μν}⧠−∂^{μ}∂^{ν}}A ^{νa } =−(1+f _{3}) g ^{2} N[A ^{ cν} A ^{ aμ} A _{ν} ^{ c }] +⋅⋅⋅+(1−s)^{2} M ^{2} A ^{ aμ}, introduced by Lowenstein to rigorously describe SU(2) Yang–Mills theory, which is written in terms of normal products. We also consider the operator product expansion A ^{ cν}(x+ξ) A ^{ aμ}(x) A ^{ bλ}(x−ξ) ∼ΣM ^{ c a bνμλ} _{ c′a′b′ν′μ′λ′ } (ξ) N[A ^{ν′c′} A ^{μ′a′} A ^{λ′b′}](x ), and using asymptotic freedom, we compute the leading behavior of the Wilson coefficients M ^{...} _{...}(ξ) with the help of a computer, and express the normal products in the field equation in terms of products of the c‐number Wilson coefficients and of operator products like A ^{ cν}(x+ξ) A ^{ aμ}(x) A ^{ bλ}(x−ξ) at separated points. Our result is −{[1+(1/α)(1+f _{4})]g ^{μν}⧠−∂^{μ}∂^{ν} }A ^{νa } =−(1+f _{3}) g ^{2}lim_{ξ→0}{ (lnξ)^{−0.28/2b }[A ^{ cν} (x+ξ) A ^{ aμ}(x) A _{ν} ^{ c }(x−ξ) +ε^{ a b c } A ^{μc }(x+ξ) ∂^{ν} A ^{ b } _{ν}(x)+⋅⋅⋅] +⋅⋅⋅}+(1−s)^{2} M ^{2} A ^{ aμ}, where β (g) =−b g ^{3}, and so (lnξ)^{−0.28/2b } is the leading behavior of the c‐number coefficient multiplying the operator products in the field equation.

Superconformal group and curved fermionic twistor space
View Description Hide DescriptionIt is shown that the superconformal transformations describe the isometry group of curved fermionic twistor space, with suitably generalized Fubini–Study Hermitian metric. Further, such a geometry is applied to derive a twodimensional supersymmetric quark–twistor string model, described by the fermionic nonlinear SU(2,2;1) invariant σmodel.

Magnetic resonances between massive and massless spin‐ 1/2 particles with magnetic moments
View Description Hide DescriptionThe general effective radial potentials for a spin‐ 1/2 particle interacting with scalar, electric, and magnetic potentials are given. In the m=0 limit, it is shown that the magnetic potential provides a well deep enough to confine the massless particle. In particular, there are exact zero‐energy solutions in which two of the four components of the massless particle are confined; only two can leak out into the asymptotic region. The scattering amplitude is analytic in the entire j plane, hence consists only of Regge poles.

Solution of boundary value problems with Laplace’s equation for ellipsoids and elliptical cylinders
View Description Hide DescriptionIf a dielectric ellipsoid or elliptical cylinder is placed in a uniform applied field it is well known that the field inside remains uniform, and is changed only by a depolarization factor that multiplies each applied field component. This paper generalizes this result. Namely, if for the three‐dimensional case the potential Φ_{app} of the applied field can be expanded in the neighborhood of the ellipsoid as Φ_{app}=J_{ l=0} ^{L}J_{ m=−l } ^{ l } D _{ l m } r ^{ l } Y _{ l m }(ϑ,φ) where l goes from zero to a maximum value L, then it is shown that the resultant potential inside the ellipsoid, Φ_{int}, is Φ_{int}=J_{ l=0} ^{ L }J_{ m=−l } ^{ l } C _{ l m } r ^{ l } Y _{ l m }(ϑ,φ) where the coefficients C _{ l m } are found explicitly and there is no C _{ l m } with l≳L. For a dielectric constant ε, the limits of the above solution as ε→∞ and ε→o are considered and are shown to yield respectively the solutions to the Dirichlet problem with potential zero on the boundary (grounded perfect conductor) and the Neumann problem with normal derivative of the potential zero on the boundary (ideal fluid flow). The homogeneous problem of free charge on an ellipsoidal perfect conductor is considered and it is shown to require a modification of the methods that yielded the results above. The modified method is applied to the problem of the oblate ellipsoid, and its limiting case of a disc, and it enables the easy derivation of various classical results due to Copson and others on the ’’problem of the electrified disc.’’ Finally, multiple dielectric ellipsoids or elliptic cylinders are considered and it is shown that problems involving such bodies can be solved in powers of a _{ i }/d _{ i j } where a _{ i } is a typical length of the ith body and d _{ i j } is the distance between the ith and the jth. This opens the way, in the proper limit of ε, to the solution of a variety of problems, such as flow around multiple strips and through the slots they may form, penetration of the electric field through perforated screens, and so on.