Volume 22, Issue 4, April 1981
Index of content:

Eigenvalue problem for an infinite tridiagonal matrix
View Description Hide DescriptionA method is developed for the calculation of the eigenvectors of an infinite tridiagonal matrix. Possible application of this method to study the problem of localization in a disordered linear chain is also discussed.

Young–tableau methods for Kronecker products of representations of the classical groups
View Description Hide DescriptionDiagrammatic methods for decomposing Kronecker products of arbitrary representations of any of the classical groups are presented. For convenience, efficient ways of computing the dimensions and quadratic Casimir’s C _{2}(R) are also given. These methods seem more useful for hand calculations than the method of Schur functions (or characteristic polynomials). An appendix presents the Kronecker products for any two representations of dimension ⩽100.

Analytical approach to initial‐value problems in nonlinear systems
View Description Hide DescriptionA method based on the equivalence between finite‐dimensional nonlinear and infinite‐dimensional linear systems of ordinary differential equations is presented in order to calculate the time‐dependent solutions of nonlinear physico‐chemical systems. The solution is cast in the form of power series whose general term is known analytically.

Laplace transforms and asymptotic expansions of orthogonal polynomials
View Description Hide DescriptionNew integral representations for orthogonal polynomials which possess a generating function are obtained by considering their Laplace transforms with respect to order. The method is used to derive some uniform asymptotic estimates for the associated Laguerre polynomials and to test an approximation scheme used in electron gas theory.

Perimeter expansion in the n‐bug system and its relationship to stability
View Description Hide DescriptionWe consider a system of n bugs located at (x _{1},y _{1}), ..., (x _{ n },y _{ n }), where bug i runs away from bug i+1 with common speed v along the instantaneous line of sight. To close the cycle of flight, bug n runs away from bug 1. The computer simulation of this system indicates that random initial configurations evolve into stable regular center‐symmetric patterns—all of which have a vertex angle of less than π/2. By utilizing the Lagrange multiplier method, we show that for these stable configurations the perimeter expansion rate ṗ is a local maximum. The most stable configuration has the smallest possible vertex angle and is associated with an absolute maximum for ṗ. The regular center‐symmetric patterns with vertex angles greater than π/2 also have a stationary perimeter expansion rate. These are local minima rather than maxima, however, and belong to configurations which are unstable.

On the Killing surface—event horizon relation
View Description Hide DescriptionA projective transformation on the scalar norm and twist of a timelike killing vector can be used to generate new space‐times. The effect of the transformation on the new Killing surface and its relation to the lcoal event horiozn is discussed. It is shown that the Geroch transformation will only connect spaces where this relation is the same.

All algebraically degenerate H spaces, via HH spaces
View Description Hide DescriptionStarting from a canonical HH space, which by virtue of the self‐dual part of its conformal curvature tensor being algebraically degenerate, has a single congruence of totally null, extremal two‐surfaces, along which the coordinates are built, we specialize to the case where the anti‐self‐dual conformal curvature vanishes, giving us an algebraically degenerate H space with coordinates especially adapted to its degeneration. We then solve the HH equation in this case and obtain, at one stroke, all algebraically degenerate H spaces in a simple, compact form, useful for later applications. The generic solution depends on four arbitrary holomorphic functions of two variables, and any special Petrov type desired is easily distinguished. Also, a special comparison is made for all such spaces of type D, showing the relation of their parameters to the usual real, type‐D parameters of mass, Newman‐Unti‐Tamburino (NUT) parameter, rotation and acceleration. Lastly, a contraction to the special cases in which the leaves of the congruence are relatively plane is performed explicitly.

The wave equation in asymptotically retarded time coordinates: Waves as simple, regular functions on a compact manifold
View Description Hide DescriptionThe Minkowski‐space scalar wave equation is represented on a spatially compact manifold with an asymptotically retarded time. In this coordinate system, with hyperbolic space slices and space coordinates generated by a conformal map, the wave equation takes a simple time‐independent form which may form a model for numerical integration calculations with nonlinear waveequations.

A further note on the Hénon–Heiles problem
View Description Hide DescriptionThe method of the Lie theory of extended groups is applied to the Hénon–Heiles problem. Only one generator for a one‐parameter group is found. The corresponding first integral is the energy. It is inferred that no other exact integral exists.

Further comments on the behavior of acceleration waves of arbitrary shape
View Description Hide DescriptionConverging waves with nonzero initial critical amplitude are completely characterized. It is shown that for a converging wave a necessary and sufficient condition for the initial critical amplitude to be zero is that the converging wave is spherical.

Functional methods in random classical field theory
View Description Hide DescriptionA functional which generatesN distinct point solutions to a classical wave equation with random coefficients in the presence of external sources is constructed. Statistical averaging over the random coefficients is then implemented using replica and/or anticommuting field techniques.

The maximality of Lorentz spaces and their importance with respect to classical and quantum electrodynamics
View Description Hide DescriptionThe indefinite metric state space S_{ M } of the covariant form of the quantized Maxwell field M contains, as is known, a family of continuously many, isomorphic, isometric pre‐Hilbert spaces L^{ q }, called Lorentz spaces, each of which corresponds to one square‐integrable, prescribed, classical, spatial distribution q(x) of the total charge Q = 0. The quotient spacesL^{ q }/N^{ q } modulo the subspaceN^{ q }⊆L^{ q } of all elements with norm 0 are indeed Hilbert spaces in S_{ M } and it appears that any QED which has to do with S_{ M } has to be formulated not on S_{ M } as a whole but on the family of these Lorentz spaces. To support this assumption we embed any L^{ q }/N^{ q } in L^{ q } in an isomorphic‐isometric way and thus get Hilbert spacesl^{ q } in S_{ M }, but not in a unique way; we will show that the different possibilities of this embedding correspond exactly with the different gauges. The main results about any embedding space l^{ q } are, however, that it is a m a x i m a lHilbert space in S_{ M } (under a premise referring to the expectation values of charge distribution), and that any Hilbert space of S_{ M } which is ’’physically important’’ in some sense is necessarily one of these l^{ q }. In this way the family {l^{ q }‖q∈Q} of Lorentz spaces (defined by the index set Q) has some outstanding properties so that the l^{ q } are now characterized by these qualities (and no longer in the heuristical way via generalization of the classical Lorentz condition). Starting now with the prominent role of the l^{ q } we get not only a deeper understanding of the Lorentz condition of classical electrodynamics—the properties of the l^{ q } lead us automatically to the definition of a positive‐definite state space of QED with the use only of these l^{ q }. (Our considerations refer not to full but to some restricted QED; the restriction is primarily given by the above‐mentioned index set Q so that extensions to full QED seem possible.) We show that our definition of a state space is consistent with time evolution, given by the Hamiltonian H, and that the QED on the basis of this state space is a constraint‐free theory because the otherwise‐necessary selection rules of Lorentz condition and charge conservation are now superfluous (and not present in a hidden form either). Furthermore, the properties of Lorentz spaces lead us automatically to a new concept of observables, all of which commute from the beginning with the operator of charge distribution. As a special observable we discuss the number operator N(k) of photons by showing that this, in general, cannot be of the form a ^{+} _{μ} a ^{μ}. A modified form of N(k) and with it a reformulation of the Hamiltonian H of QED is given. All these considerations go back to the properties of the Lorentz spaces and thus, basically, to the canonical quantization of the Maxwell field.

Phase‐space approach to relativistic quantum mechanics. III. Quantization, relativity, localization and gauge freedom
View Description Hide DescriptionWe examine the relationship between the mathematical structures of classical mechanics,quantum mechanics, and special relativity, with a view toward building a consistent framework for all three. The usual idea of ’’canonical quantization,’’ with its emphasis on the transition from functions over classical phase space to operators, appears to be inconsistent with relativistic covariance. On the other hand, the spectral condition of relativistic quantum mechanics, which is merely a covariant statement that the energy is nonnegative, naturally leads to the construction of a covariant extension of classical phase space. By giving up the idea of sharp ’’localizability’’ in space—known to be at odds with covariance— and adopting instead a notion of ’’soft’’ localizability in phase space, a consistent theory of relativistic quantum mechanics is seen to emerge whose structure naturally incorporates the classical sympletic geometry as well. Furthermore, the new theory deals directly and covariantly with extended particles rather than point particles and is free of the various inconsistencies known to plague the usual theory. The new notion of ’’microlocality’’ in phase space leads to a new form of gauge freedom which is similar to the usual one but simpler and more powerful, using methods of complex analysis. The phase‐space version of Yang–Mills theory is worked out.

Path integrals with a periodic constraint: The Aharonov–Bohm effect
View Description Hide DescriptionThe Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.

A new set of coherent states for the isotropic harmonic oscillator: Coherent angular momentum states
View Description Hide DescriptionThe Hamiltonian for the oscillator has earlier been written in the form where ν^{†} and ν are raising and lowering operators for ν^{†}ν, which has eigenvaluesk (the ’’radial’’ quantum number), and λ^{†} and λ are raising and lowering 3‐vector operators for λ^{†}⋅λ, which has eigenvaluesl (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L^{2}, and L _{3} (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N _{1}, N _{2}, and N _{3}. It is proposed to call them coherent angular momentum states. They are shown to be minimum‐uncertainty states for the variables ν, ν^{†} , λ, and λ^{†}, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K_{3}, where K_{3} is the three‐dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.

Iteration of single‐ and two‐channel Schrödinger equations
View Description Hide DescriptionA general perturbation technique is developed for the iteration of one‐ and two‐channel Schrödinger equations for potentials, which can be expanded around a minimum. The channel coupling is assumed to be weak. In order to facilitate the numerical or algebraic computer calculation of terms of higher order, a recurrence relation is derived for some particularly important coefficients. The eigenvalues and oscillatorlike solutions are then derived explicitly up to and including the third‐order iteration. Furthermore, we demonstrate that certain parts of every nth order iteration can be lumped together in a manner which is independent of the specific form of the potential. Finally, these methods are applied to the calculation of the eigenvalues of the one‐channel equation for linear and logarithmic potentials with or without a weak Coulomb contribution.

Derivation of ’’Bethe’s hypothesis’’ from the quantum inverse scattering transformation for the nonlinear Schrödinger equation
View Description Hide DescriptionUsing the quantum version of the inverse scattering transformation for the nonlinear Schrödinger equation, eigenstates of the Hamiltonian can be constructed. We show that these eigenstates are of the Bethe form.

Hypervirial relations and orthogonalization conditions
View Description Hide DescriptionThrough the imposition of orthogonalization conditions, a number of results are derived for wavefunctions and eigenvalues in the framework of hypervirial methodology. Some previous results are extended and various similar developments of off‐diagonal Hellmann–Feynman and sum rule formulas are indicated.

A semiclassical treatment of path integrals for the spin system
View Description Hide DescriptionStarting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase‐space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.

Semiclassical approximations at positive temperatures in stochastic physics
View Description Hide DescriptionSemiclassical approximations are developed for stochastic mechanics and stochastic field theory at positive temperatures. The tunneling phenomena of Euclidean quantum mechanics is seen to have a statistical interpretation. A semiclassical algorithm for calculating the generating functional of the moments of the positive‐temperature process is developed. Positive‐temperature fluctuations around a scalar soliton and in the pure SU(2) Yang–Mills theory are also briefly considered.