Volume 15, Issue 12, December 1974

Weyl conform tensor for stationary gravitational fields
View Description Hide DescriptionOur formulas for the Weyl conform tensor components generalize results published earlier by Z. Perjés for vacuum fields. We also offer an abstract version of these equations which may shed some light upon their structure. The expressions for the Weyl conform tensor are specialized to the case of small perturbations from a stationary axially symmetric background geometry. The resulting formulas supplement the expressions which Chandrasekhar and Friedman have developed for the components of the Ricci tensor. We anticipate that this will facilitate the comparison of the CF perturbation theory with the recent studies of perturbations of the Kerr metric by Press, Teukolsky, and Wald. In this connection we identify in terms of the CF field variables the fields which are involved in Teukolsky's separable field equations.

On irreducible corepresentations of finite magnetic groups
View Description Hide DescriptionWe have obtained a set of homogeneous linear equations in the Clebsch‐Gordan coefficients for the Kronecker inner direct product of two irreducible corepresentations of a finite magnetic group. The solutions of these equations give the Clebsch‐Gordan coefficients even when the group is not simply reducible. The nontrivial Clebsch‐Gordan coefficients for the magnetic group C _{4ν}(C _{2ν}) have been evaluated. We have also investigated the criterion determining whether a particular irreducible corepresentation is equivalent to its complex conjugate representation. A projection operator has been constructed for obtaining the basis pertaining to a particular irreducible corepresentation.

Dynamics of a multilevel Wigner‐Weisskopf atom
View Description Hide DescriptionWe study the dynamics of an atom with a finite number of discrete energy levels weakly coupled to a continuum of energy levels, showing that any bound state undergoes a decay into the continuum which, in the limit as the coupling constant goes to zero, becomes rigorously exponential.

Clebsch‐Gordan coefficients and special functions related to the Euclidean group in three‐space
View Description Hide DescriptionIn this paper the Clebsch‐Gordan coefficients of the Euclidean group in 3‐space are explicitly and rigorously determined. The results are used to give elegant derivations of identities involving Wigner D functions and spinor functions.

Scattering theory for Schrödinger operators with L ^{∞} potentials and distorted Bloch waves
View Description Hide DescriptionWe prove that, if and are real‐valued functions, the wave operators associated with the self‐adjoint operators H _{1}=−Δ+q _{1} and H _{2}=−Δ+q _{1}+q _{2} in L ^{2}(R ^{3}) exist and are complete. We also prove that, if q _{1} is periodic and q _{2} is in a certain weighted L ^{2} , the absolutely continuous part of H _{2} possesses two sets of generalized eigenfunctions which belong to the dual of and are solutions of linear equations involving the generalized eigenfunctions of H _{1}.

Relativistic quantum mechanics and local gauge symmetry
View Description Hide DescriptionThe requirement that (either Abelian or non‐Abelian) local symmetry transformations be globally and unitarily implementable kinematical symmetries of relativistic systems implies the emergence of a dynamical group which has been suggested in earlier studies. The group leads to a 4‐velocity operator and to the Newton‐Wigner position operator. Demanding gauge invariance of localization determines a unique interaction structure. Superselection rules for the gauge charges arise.

Asymptotic solutions of second‐order linear equations with three transition points
View Description Hide DescriptionA uniformly valid asymptotic expansion is obtained for the regular solution of a class of second‐order linear differential equations with three transition points‐a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.

A geometric interpretation of classical relativistic electrodynamics
View Description Hide DescriptionA solution is offered for this problem: Describe the observables of classical electrodynamics with connections on fiber bundles without using nonobservable entities, either in computations or in conceptual development. The solution employs a connection on the affine frame bundle of space‐time. Comparisons are made with other geometric interpretations of electrodynamics.

Summation of regularized perturbative expansions for singular interactions
View Description Hide DescriptionIn this paper we give a first application of a general method whose mathematical aspects will be fully developed in a forthcoming article. We are concerned with strongly singular perturbative series. Here we shall restrict ourselves to the most general two‐body repulsive singular potential for which a regularization exists. Various extensions of this case are discussed in the conclusion. We show that, knowing only a finite number of regularized Born terms, it is possible to construct an upper bound to the exact phase shifts and that this upper bound is the best possible for the given regularization. The method uses the construction of the [N/N] Padé approximation indifferently on the regularized partial waves of the K or T matrix and exploits the fact that the approximate corresponding phase shifts have an absolute minimum as a function of the regularization parameter (cutoff). Three numerical examples are provided which show, even for very large phase shifts, an excellent convergence.

Phase properties of some photon states with nonzero energy density
View Description Hide DescriptionWe describe some photon states with nonzero energy density in the whole space; these states are obtained by taking a finite number of photons within a finite box and letting the volume and the number of photons go to infinity according to usual procedure in statistical mechanics. In such a limit we describe an observable phase operator; we investigate its properties both in the case of free field and in the case of coupling with prescribed classical sources. Finally we give a quantum description of uniform static field.

The semiclassical fermion μ‐space density in three dimensions
View Description Hide DescriptionAn approximation is constructed to the phase‐space, or Wigner, distribution function for a three‐dimensional, dense Fermi gas in a spherically symmetric potential well. Near the surface separating the classically forbidden region from the classically allowed region, quantum oscillations occur. The oscillations are expressed in terms of a universal function.

Baker‐Campbell‐Hausdorff formulas
View Description Hide DescriptionBaker‐Campbell‐Hausdorff formulas can be constructed simply by matrix multiplication. Examples are given.

On the existence of weakly retarded and advanced Green's functions
View Description Hide DescriptionBy considering a model field equation that contains the acausal propagation features of a spin 3/2 field we show that depending on the ``external field'' one can either have weakly retarded fundamental solutions or not.

The algebra and group deformations , , and
View Description Hide DescriptionWe discuss a class of deformations of the inhomogeneous classical algebras to k(n,m) for 1 ≤ m ≤ n. This generalizes the previously known expansions i k(n)⇒k(n,1). As the title indicates, this is done explicitly for the orthogonal, unitary, and symplectic cases. We construct the corresponding deformed groups K(n,m) as multiplier representations on the space of functions over the rank m coset space K(n ‐ m)\K(n). This method allows us to build a principal series of unitary representations of K(n,m). The contractions of the deformed algebras and groups are considered.

Canonical transforms. II. Complex radial transforms
View Description Hide DescriptionContinuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n‐dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut‐Girardello, one‐dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker‐Campbell‐Hausdorff and special function relations.

Green's function for Laplace's equation in an infinite cylindrical cell
View Description Hide DescriptionThe Green's function for Laplace's equation in an infinite‐length cylinder with a homogeneous mixed boundary condition is considered. Its eigenfunction expansion converges slowly when the axial separation between the source and observation points is small compared to the cylinder radius, and diverges when the axial separation is zero. Applying a modified form of a contour integral method of Watson to an integral representation of the Green's function, a more general expansion of the Green's function is derived. Watson's original method had previously been applied to the case when the source and observation points were both on the axis of the cylinder. The expansion contains a free parameter which may be adjusted to give rapid convergence for any axial separation. It fails, however, when the source and observation points are both near the surface of the cylinder. For two special values of the parameter, the general expansion reduces to the eigenfunction expansion or to the integral representation. The derivation is somewhat obscure, but the resulting formula has a simple interpretation as the superposition of the potential of two related boundary value problems in finite‐length cylinders. Some numerical results are given in the spatial region which previously could not be calculated, for a boundary condition approaching a homogeneous Neumann condition, and for a homogeneous Dirichlet condition.

Limits of the Tomimatsu‐Sato gravitational field
View Description Hide DescriptionThe Tomimatsu‐Sato (TS) solutions of the Einstein field equations are studied in several limiting cases. In the weak‐field limit we construct two Newtonian models for the source, one consisting of a rotating disc of radius a/n, the other made up of n complex point multipoles. The ``extreme'' limit q = 1 is also examined in detail, and we find there are many distinct ways of taking this limit. We are thereby led to a new two‐parameter family of exact solutions which, unlike the TS metrics, are not asymptotically flat.

Effects of long range interactions in harmonically coupled systems. I. Equilibrium fluctuations and diffusion
View Description Hide DescriptionSystems of harmonically coupled identical particles at thermal equilibrium provide dynamical models for studies of diffusion due to equilibrium fluctuations. The velocity autocorrelation function and mean square displacement of a particle selected from a given system are investigated for various models which have the common feature that the particle is directly coupled to L > 1 neighbors, reflecting the influence of long range interactions. Theorems are developed which indicate how the time course of diffusion is dictated by analytic properties of the vibrational frequency distribution as well as by quantum fluctuations whose presence is betrayed by the increasingly important role at progressively lower temperatures of τ_{ q } = ℏ/πk T, the quantum transient time. The formalism is first applied to a system for which the long range couplings are so parametrized by a range parameter z that when z=0 the frequency distribution is identical to that for nearest neighbor coupling only (L=1), while as z approaches unity (L→∞) the frequency distribution becomes identifiable with that of Ford, Kac, and Mazur which served as the starting point for their dynamical theory of Brownian motion. Consequences of this model are: (1) when z <0.5, the classical velocity autocorrelation functions exhibit similar qualitative features to those computed for molecular diffusion in simple liquids; (2) as z approaches unity, the classical velocity autocorrelation function approaches the e ^{‐λτ} Gaussian Markoffian form, and the mean square displacement in the same limit is identical to that predicted by the Langevin equation; (3) at low temperatures such that λτ_{ q }>1, quantum fluctuations tend to dominate thermal fluctuations, resulting in severe departures from Gaussian Markoffian behavior. The low temperature effects are analyzed in some detail, and it is suggested that the predicted departure of the mean square displacement from its classical behavior might be displayed by a particle of macroscopic size suspended in a superfluid. Other models are developed which yield mean square displacements which depart even at high temperature from the linear dependence upon time characteristic of classical diffusion. The reasons and possible physical implications of these behaviors are discussed, together with a brief consideration of Poincaré cycles, whose neglect is implicit in any dynamical theory of irreversible processes.

Summation relation for U(N) Racah coefficients
View Description Hide DescriptionA summation relation is given for U(N) Racah coefficients which has the form of an orthogonality relation, or a composition of recoupling transformations, except that the summation over column indices (for fixed row indices) is over multiplicity labels only. In the recoupling matrix for [f ^{1}] × [f ^{2}] × [f ^{3}] → [f], U(N) irreducible representations [f ^{2}] and [f ^{3}] are limited to be elementary, [11…10…0]≡[1^{ k }], or totally symmetric [k], or of the form [k ^{ N−1}]. Results are tabulated as functions of the axial distances in [f] for [f ^{2}]=[1^{ N−1}], [1^{ N−2}], or [2^{ N−1}]; [f ^{3}]=[1], [1^{2}], or [2]; all cases which arise in the evaluation of squares of matrix elements of one‐ and two‐body operators averaged over irreducible representations of U(N).

Bäcklund transformations for certain nonlinear evolution equations
View Description Hide DescriptionBäcklund transformations associated with the Korteweg‐deVries (KdV), modified KdV, and nonlinear Schrödinger equations are derived by a method due to Clairin. Also, a Bäcklund transformation relating the KdV and modified KdV equations is obtained by the same technique.