Volume 16, Issue 9, September 1975
Index of content:

Orthonormality properties of double coset matrix elements
View Description Hide DescriptionThe orthogonality and completeness conditions of group representation theory are shown to provide complementary orthonormality relations for weighted double coset matrix elements. The double coset matrix elements are appropriate for symmetry adaption of the basis to two subgroup sequences using the double coset decomposition H‐G/K. When applied to the symmetric and unitary groups, the double coset matrix elements assume the role of recoupling coefficients. The orthonormality properties give nontrivial relations between coefficients coupling tensors of different rank and/or different dimensions.

Generalized P‐representation for bounded operators
View Description Hide DescriptionWe generalize the notion of P‐representation, introducing a space of generalized functions, such that every Hilbert–Schmidt operator has a P‐representation. Then we extend that notion to another space of generalized function, such that every bounded operator has a P‐representation.

Jacobi’s principle for the case of time‐varying Hamiltonian
View Description Hide DescriptionWe present a generalization of Jacobi’s principle to the case where the particle Hamiltonian varies with time, illustrating the formalism with an example from particle mechanics. It can be applied also to Stueckelberg mechanics to the case of variable mass, and we give an example.

On the solution of the phase retrieval problem
View Description Hide DescriptionIt is shown that the intensity in the image plane of a microscope determines u n i q u e l y the phase of the corresponding image wavefunction up to an over‐all phase. This result is obtained using the a p r i o r i information that both the image wavefunction and the unperturbed wavefunction in the Fraunhofer plane are band‐limited and that we have some a p r i o r i knowledge about the intensity at the rim of the diaphragm in the Fraunhofer plane. If we have no useful a p r i o r i information about the wavefunction in the Fraunhofer plane, unique phase reconstruction is possible from two exposures, corresponding to two different values of the defocusing.

Remarks on the Green’s functions for the cubic lattices
View Description Hide DescriptionA simpler way of finding the Green’s functionG (2p,0,0) for the body‐centered and face‐centered cubic lattices as the square and product, respectively, of the pth derivatives of two complete elliptic integrals of the first kind is pointed out. The key relations required for these results are the Clausen’s and Brafman’s formulas for the body‐centered and face‐centered cubic lattices, respectively.

Canonical transformations and path integrals
View Description Hide DescriptionA limited class of canonical transformations is introduced into the Lagrangianpath integral method of quantization. Path integral quantization in different representations is discussed and a simple example is given.

A variational principle for the Boltzmann equation for hot electrons in a semiconductor
View Description Hide DescriptionIt is shown that the well‐known Kohler variational principle for the small electric‐field solution of the Boltzmann equation can be extended to arbitrary fields, if a generalization of Hamilton’s principle proposed by Djukic and Vujanovic is used.

On Ursell’s combinatorial problem
View Description Hide DescriptionA combinatorial problem considered by H. D. Ursell in his seminal paper on cluster theory [Proc. Camb. Phil. Soc. 23, 685 (1927)] is studied. Ursell’s analysis, which is not rigorous, is described by Fowler and Guggenheim as being far from simple. In this paper we arrive at Ursell’s result using a method which is straightforward, yet completely rigorous.

Discrete space quantum mechanics
View Description Hide DescriptionThe general problem of finding exactly soluble quantum systems is considered. It is argued that discrete space quantum mechanics emerges in a natural way as an avenue of approach. Discrete space quantum mechanics is formulated and applied to one‐dimensional quantum systems with emphasis on single‐channel models. It is found that a large variety of systems are exactly soluble in the sense that they only require the inversion of a finite‐dimensional matrix. The interaction may in general be both nonlocal and non‐time‐reversal‐invariant. The analytic structure of the resolvent is worked out in detail for a simple class of examples. It is shown that a slightly modified version of the usual continuous space Schrödinger equation may in principle be solved exactly for any finite range local potential by writing the solution in terms of corresponding discrete space solutions. It is also shown that from an algebraic viewpoint the models constructed are realizations of generalized versions of the Weyl relations.

Ray‐optical analysis of fields on shadow boundaries of two parallel plates
View Description Hide DescriptionThe electromagnetic diffraction by two parallel plates of semi‐infinite length is treated by ray methods. Two special problems are considered: (i) calculation of the fields in the forward and backward directions due to diffraction of a normally incident plane wave by two nonstaggered parallel plates; (ii) calculation of the field due to a line source in the presence of two staggered parallel plates when the source, the two edges, and the observation point are on a straight line. The crucial step in the ray‐optical analysis is the calculation of the interaction between the plates. This calculation is performed by two methods, namely, the uniform asymptotic theory of edge diffraction and the method of modified diffraction coefficient. The relative merits of the two methods are discussed. The ray‐optical solution of problem (i) agrees with the asymptotic expansion (plate separation large compared to wavelength) of the exact solution.

Gauge dependence of Green’s functions in quantum electrodynamics from parallel translation
View Description Hide DescriptionThe classical concept of parallel translation is extended to scalar quantum electrodynamics in order to give a gauge‐independent definition of differentiation. This is achieved by a suitable definition of time ordering for operator products. However, due to some essentially nonlocal commutation relations, the differential equations for the Green’s functions are still gauge‐dependent. The gauge dependence of the commutators can be removed by parallelism at large. Since this is not considered to be a physically reasonable concept, the gauge dependence of the Green’s functions is discussed for general linear gauges in space–time.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. VI. Comment on the notion of a cell material
View Description Hide DescriptionThe concept of a cell material introduced by Miller is reinvestigated in connection with Brown’s assertion that an asymmetric cell material is not self‐consistent [J. Math. Phys. 15, 1516 (1974)]. We construct a simple example of the asymmetric cell material which is in fact self‐consistent. The misleading interpretation of the asymmetric cell material is due to Miller rather than to Brown.

Operator algebra of dual resonance models
View Description Hide DescriptionProperties of a set of operators introduced by Baker, Coon, and Yu are discussed. The operators involve generalizations of harmonic oscillator operators and facilitate the construction of a family of dual resonance models which includes the Veneziano model as a limiting case. Matrix representations of the operators are constructed, and it is shown that the operators have finite norm in contrast with the unboundedness of creation and annihilation operators of the Veneziano model.

Null geodesic deviation. I. Conformally flat space–times
View Description Hide DescriptionThe equation of geodesic deviation is solved in conformally flat space–time in a covariant manner. The solution is given as an integral equation for general geodesics. The solution is then used to evaluate second derivatives of the world function and derivatives of the parallel propagator, which need to be known in order to find the Green’s function for wave equations in curved space–time. A method of null geodesic limits of two‐point functions is discussed, and used to find the scalar Green’s function as an iterative series.

Propagation of high frequency surface waves along cylinders of general cross section
View Description Hide DescriptionThe propagation of high frequency scalar surface waves along the generators of a homogeneous cylinder which has a cross‐sectional boundary of nonconstant curvature is investigated. Asymptotic solutions are obtained to the reduced wave equation, subject to an impedance boundary condition at the surface of the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature, and the higher order modes have nulls. The case of closed boundary curves is also discussed. In a companion joint paper by L. O. Wilson and the author, the asymptotic procedures developed for the analysis of the scalar problem will be applied to the investigation of the propagation of elasticsurface waves(Rayleigh waves) along a homogeneous isotropic cylinder with stress‐free boundary. This problem arises in connection with guided acoustic surface waves.

Propagation of high frequency elastic surface waves along cylinders of general cross section
View Description Hide DescriptionThe propagation of high frequency elasticsurface waves along the generators of a homogeneous isotropic cylinder which has a cross‐sectional boundary of nonconstant curvature is investigated. The boundary surface is stress‐free and the surface waves, or Rayleigh waves, are disturbances whose amplitudes decay rapidly with depth into the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature. The application of these high frequency asymptotic results to the case of a closed boundary curve is discussed. Particular cases will be investigated in more detail in a subsequent paper.

All stationary axisymmetric rotating dust metrics
View Description Hide DescriptionThe Einstein equations for stationary axisymmetric space–times with a rotating dust source are systematically reduced to quadratures. The general solution depends upon an arbitrary axisymmetric solution of the flat three‐dimensional Laplace equation and upon an arbitrary function of one variable.

Correlation inequalities for two‐dimensional vector spin systems
View Description Hide DescriptionA set of correlation functioninequalities including Griffiths, Kelly, Sherman type inequalities are proven for a lattice system of N sites where on each site there is a vector spin [s= (s ^{ x },s ^{ z }), ‖s‖=1] whose distribution of values over the unit circle is given by f (s), where f (s) =f (−s). The spins interact through two‐body, anisotropic,ferromagnetic interactions. Also an external field h, h ^{ x }⩾0, and h ^{ z }⩾0, is present. The proof uses Gaussian random variables.

Dilatations and the Poincaré group
View Description Hide DescriptionWe discuss the projective unitary representations of the Weyl group (Poincaré group enlarged with dilatations).

Properties of linear representations with a highest weight for the semisimple Lie algebras
View Description Hide DescriptionA theory for the representations with a highest and/or lowest weight is given for the semisimple complex Lie algebras (and their real forms). These representations are either irreducible finite‐dimensional, irreducible infinite‐dimensional or reducible, but not completely reducible, infinite‐dimensional (called elementary representations), depending upon the property of the associated highest (or lowest) weight Λ. No restriction is made to those representations of the semisimple Lie algebras which can be integrated to form representations of the corresponding Lie group. The algebraA _{1} is chosen (Sec. III) as a simple and familiar example upon which, however, much of the proof for the results obtained for the theory of representations with a highest (and/or lowest) weight for the general case of a semisimple Lie algebra rests (Sec. IV). It is demonstrated that the irreducible representations D (Λ) with a highest (and/or lowest) weight Λ of the semisimple Lie algebras decompose with respect to any (regularly) embedded subalgebra of the type A _{1} in the manner that either (a) the subrepresentations subduced on A _{1} are all irreducible finite‐dimensional, or (b) all infinite‐dimensional. If for case (b) the complex number M ^{α} ≡ 2(M,α)/(α,α), α the (simple) root of A _{1} and M a weight of D (Λ) extremal with respect to A _{1}, is not a nonnegative integer, then the representation subduced on A _{1} is irreducible. If, however, M ^{α} is a nonnegative integer, then a reducible but not completely reducible, representation is subduced on A _{1}. Based upon the results of Sec. IV a generalization of Freudenthal’s formula is obtained in Sec. V, valid for irreducible infinite‐dimensional representations with highest (or lowest) weight. In Sec. VI generalizations are given of Racah’s recurrence relation for the multiplicity of weights, Weyl’s character formula and Kostant’s formula for the multiplicity of weights for infinite‐dimensional irreducible representations with a highest (or lowest) weight of the semisimple Lie algebras. These formulas are derived utilizing theorems and lemmas obtained by Verma, I. M. Gel’fand, S. I. Gel’fand, Bernstein, Harish‐Chandra and the results of Sec. IV. In Sec. VII some of the infinite‐dimensional representations of the algebraA _{2} are discussed as examples, employing the geometrical methods developed by Antoine and Speiser and by Biedenharn and others.