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Dec 1974

Volume 15, Issue 12, pp. 2027-2257

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Weyl conform tensor for stationary gravitational fields

Frederick J. Ernst

J. Math. Phys. 15, 2027 (1974); http://dx.doi.org/10.1063/1.1666576 (4 pages) | Cited 3 times

Online Publication Date: 4 November 2003

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Our 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

P. Rudra

J. Math. Phys. 15, 2031 (1974); http://dx.doi.org/10.1063/1.1666577 (5 pages) | Cited 15 times

Online Publication Date: 4 November 2003

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We 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 C4ν(C2ν) 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

E. B. Davies

J. Math. Phys. 15, 2036 (1974); http://dx.doi.org/10.1063/1.1666578 (6 pages) | Cited 22 times

Online Publication Date: 4 November 2003

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We 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

Jung Sik Rno

J. Math. Phys. 15, 2042 (1974); http://dx.doi.org/10.1063/1.1666579 (6 pages) | Cited 2 times

Online Publication Date: 4 November 2003

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In 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

Giovanni M. Troianiello

J. Math. Phys. 15, 2048 (1974); http://dx.doi.org/10.1063/1.1666580 (5 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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We prove that, if q1 ϵ C0(R3)∩L∞(R3) and q2 ϵ L1(R3)∩L2(R3) are real‐valued functions, the wave operators associated with the self‐adjoint operators H1=−Δ+q1 and H2=−Δ+q1+q2 in L2(R3) exist and are complete. We also prove that, if q1 is periodic and q2 is in a certain weighted L2 space X, the absolutely continuous part of H2 possesses two sets of generalized eigenfunctions which belong to the dual space X* of X and are solutions of linear equations involving the generalized eigenfunctions of H1.

Relativistic quantum mechanics and local gauge symmetry

P. Roman and J. P. Leveille

J. Math. Phys. 15, 2053 (1974); http://dx.doi.org/10.1063/1.1666581 (10 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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The 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

Ali H. Nayfeh

J. Math. Phys. 15, 2063 (1974); http://dx.doi.org/10.1063/1.1666582 (4 pages) | Cited 1 time

Online Publication Date: 4 November 2003

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A 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

Richard Petti

J. Math. Phys. 15, 2067 (1974); http://dx.doi.org/10.1063/1.1666583 (4 pages)

Online Publication Date: 4 November 2003

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A 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

D. Bessis, L. Epele, and M. Villani

J. Math. Phys. 15, 2071 (1974); http://dx.doi.org/10.1063/1.1666584 (8 pages) | Cited 16 times

Online Publication Date: 4 November 2003

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In 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

J. P. Provost, F. Rocca, G. Vallee, and M. Sirugue

J. Math. Phys. 15, 2079 (1974); http://dx.doi.org/10.1063/1.1666585 (7 pages) | Cited 5 times

Online Publication Date: 4 November 2003

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We 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

N. L. Balazs and G. G. Zipfel

J. Math. Phys. 15, 2086 (1974); http://dx.doi.org/10.1063/1.1666586 (4 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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An 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

R. Gilmore

J. Math. Phys. 15, 2090 (1974); http://dx.doi.org/10.1063/1.1666587 (3 pages) | Cited 63 times

Online Publication Date: 4 November 2003

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

On the existence of weakly retarded and advanced Green's functions

Anton Z. Capri

J. Math. Phys. 15, 2093 (1974); http://dx.doi.org/10.1063/1.1666588 (3 pages) | Cited 1 time

Online Publication Date: 4 November 2003

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By 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 Im [SO(n)⊗SO(m)]⇒SO(n,m), Im [U(n)⊗U(m)]⇒U(n,m), and Im [Sp(n)⊗Sp(m)]⇒Sp(n,m) for 1 ⩽ m ⩽ n

Kurt Bernardo Wolf and Charles P. Boyer

J. Math. Phys. 15, 2096 (1974); http://dx.doi.org/10.1063/1.1666589 (6 pages) | Cited 14 times

Online Publication Date: 4 November 2003

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We discuss a class of deformations of the inhomogeneous classical algebras im [k(n)⊕k(m)] 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

Kurt Bernardo Wolf

J. Math. Phys. 15, 2102 (1974); http://dx.doi.org/10.1063/1.1666590 (10 pages) | Cited 19 times

Online Publication Date: 4 November 2003

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Continuing 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 Lrn−12(0,∞) 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 SL(2,math) 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

A. Peskoff

J. Math. Phys. 15, 2112 (1974); http://dx.doi.org/10.1063/1.1666591 (9 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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The 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

William Kinnersley and Edward F. Kelley

J. Math. Phys. 15, 2121 (1974); http://dx.doi.org/10.1063/1.1666592 (6 pages) | Cited 14 times

Online Publication Date: 4 November 2003

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The 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

Paul E. Phillipson

J. Math. Phys. 15, 2127 (1974); http://dx.doi.org/10.1063/1.1666593 (21 pages) | Cited 11 times

Online Publication Date: 4 November 2003

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Systems 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 τq = 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 λτq>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

K. T. Hecht

J. Math. Phys. 15, 2148 (1974); http://dx.doi.org/10.1063/1.1666594 (9 pages) | Cited 10 times

Online Publication Date: 4 November 2003

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A 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 [f1] × [f2] × [f3] → [f], U(N) irreducible representations [f2] and [f3] are limited to be elementary, [11…10…0]≡[1k], or totally symmetric [k], or of the form [kN−1]. Results are tabulated as functions of the axial distances in [f] for [f2]=[1N−1], [1N−2], or [2N−1]; [f3]=[1], [12], 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

G. L. Lamb

J. Math. Phys. 15, 2157 (1974); http://dx.doi.org/10.1063/1.1666595 (9 pages) | Cited 77 times

Online Publication Date: 4 November 2003

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Bäcklund 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.

Orthogonal polynomials from the viewpoint of scattering theory

K. M. Case

J. Math. Phys. 15, 2166 (1974); http://dx.doi.org/10.1063/1.1666597 (9 pages) | Cited 23 times

Online Publication Date: 4 November 2003

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It is demonstrated that there is a close parallel between the theory of a class of orthogonal polynomials and scattering theory. In both cases a fundamental role is played by a particular solution of the basic difference (differential) equation which we call the Jost function. Under rather general conditions this function has simple analytic properties. It determines and is largely determined by either the asymptotic phases or the continuous part of the weight (spectral) function. Indeed this is more than an analogy. By appropriate limiting procedures one can pass from a result about orthogonal polynomials to one in scattering theory. Conversely, scattering theory throws considerable light on theorems about orthogonal polynomials.

A remark on the Green's function for the face‐centered cubic lattices

Koichi Mano

J. Math. Phys. 15, 2175 (1974); http://dx.doi.org/10.1063/1.1666598 (2 pages) | Cited 5 times

Online Publication Date: 4 November 2003

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The lattice Green's function G(2p,0,0) for the face‐centered cubic lattices, which was obtained by Inoue as a linear combination of the F4 function of Appell, is shown to be expressible as the product of the pth derivatives of two complete elliptic integrals of the first kind.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. III. Perturbation treatment of the effective permittivity in completely random heterogeneous materials

Motoo Hori and Fumiko Yonezawa

J. Math. Phys. 15, 2177 (1974); http://dx.doi.org/10.1063/1.1666599 (9 pages) | Cited 23 times

Online Publication Date: 4 November 2003

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Perturbation expansion series are derived for the effective permittivity of completely random heterogeneous materials. The formulation is performed by regarding a completely random material as a limiting case of an isotropic cell material. It is emphasized that, in order to obtain a physically reasonable and mathematically correct result, the ``exclusion effect'' must be taken into account in the averaging procedure. Prescription for evaluating the perturbation coefficient of an arbitrary order is given and explicit forms of leading terms are presented. The results bear a wide variety of applications in calculating effective physical constants such as dielectric constant, magnetic permeability, electrical and thermal conductivity, and diffusion constant. It is mentioned that the idea and formulation in this article are important for more general approximations (to be studied in the succeeding Paper IV) especially in connection with the problem of electron localization in some disordered systems.

Power statistics for wave propagation in one‐dimension and comparison with radiative transport theory. II

W. Kohler and G. C. Papanicolaou

J. Math. Phys. 15, 2186 (1974); http://dx.doi.org/10.1063/1.1666600 (12 pages) | Cited 30 times

Online Publication Date: 4 November 2003

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We consider the one‐dimensional problem of a slab having a random index of refraction and illuminated from within by a point source. We compute the expected value and the fluctuations of both the total power and power flux. These quantities, which are functions of the slab width, source location, and observation point, are determined in the limit of weak refractive index fluctuations and large slab thickness. We compare the expected values of total intensity and flux with the predictions of radiative transport theory. We also compare the results of both theories with numerical simulations.

Proof of the charge superselection rule in local relativistic quantum field theory

F. Strocchi and A. S. Wightman

J. Math. Phys. 15, 2198 (1974); http://dx.doi.org/10.1063/1.1666601 (27 pages) | Cited 173 times

Online Publication Date: 4 November 2003

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The paper interprets and proves the charge superselection rule within the framework of local relativistic field theory as the statement that the charge operator commutes with all quasilocal observables. Once the basic formalism expressing the property of locality of the observables has been accepted, the proof is an elementary application of Gauss law relating the electric charge in a region to the flux of electric field through the boundary of the region. Most of the paper is devoted to the evidence that the indefinite metric formalism and its accompanying definitions of gauge, gauge transformation, and gauge invariance are internally coherent and consistent with the evidence from free field theory and the renormalized perturbation theory of coupled fields. The paper closes with speculations on analogous explanations of the baryon and lepton superselection rules within the framework of gauge models of strong and weak interactions.
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