Volume 13, Issue 4, April 1972
Index of content:

The Angular Spectrum Representation of Electromagnetic Fields in Crystals. I. Uniaxial Crystals
View Description Hide DescriptionThe electromagnetic field in a linear, nonmagnetic, nonabsorbing uniaxial crystal which fills the entire half‐space z ≥ 0 and whose optic axis is perpendicular to the plane z = 0 is represented as an angular spectrum of plane waves. The angular spectrum representation consists of a superposition of plane electromagneticwaves expressed as the sum of two integrals. In general both homogeneous and evanescent plane waves are required in each integral. Each plane wave of the spectrum satisfies the identical equations obeyed by the entire field. The homogeneous plane waves of the first integral are all ordinary waves and those of the second integral all extraordinary waves. The spectral amplitudes of the field are explicitly expressed in terms of the Fourier transform of the field in the place z = 0. The method of stationary phase is applied to the integral representation and it is thereby shown that in the far zone the field may be expressed as the sum of an outgoing (nonuniform) spherical wave and an outgoing (nonuniform) ellipsoidal wave. The amplitude of these waves, at each point on the wave surface, is expressed in terms of the Fourier transform of the field in the plane z = 0.

The Angular Spectrum Representation of Electromagnetic Fields in Crystals. II. Biaxial Crystals
View Description Hide DescriptionThe electromagnetic field in a linear, nonmagnetic nonabsorbing biaxial crystal which fills the entire half‐space z ≥ 0 and which has one of its principal dielectric axes perpendicular to the plane z = 0 is represented as an angular spectrum of plane waves. The angular spectrum representation consists of a superposition of plane waves expressed as the sum of two integrals. Each integral contains in general both homogeneous and evanescent plane waves. Each plane wave of the angular spectrum (whether homogeneous or evanescent) satisfies the identical equations which are obeyed by the entire field. The spectral amplitudes of the field are explicitly expressed in terms of the Fourier transform of the field in the plane z = 0. The special case of a uniaxial crystal whose optic axis is parallel to the plane z = 0 is treated in some detail. The far zone structure of the field in such a crystal is determined using the method of stationary phase. The field in the far zone is expressed explicitly in terms of the Fourier transform of the field in the plane z = 0.

An Analytical Approach to the Theory of Internal Conical Refraction
View Description Hide DescriptionAn analytical, quantitative treatment of internal conical refraction is presented. The recently derived angular spectrum representation of electromagnetic fields in biaxial crystals is employed. Explicit expressions for the electric field in the far zone of a conically refracted beam of light are found by applying the principle of stationary phase to the integrals of the representation. A close examination of the expressions for the field in the far zone yields all the known results of internal conical refraction including the Poggendorf dark band. The present treatment, however, goes beyond previous studies since, with our expressions, it is possible to investigate the detailed structure of the conically refracted field.

Internal Labeling and the Group G _{2}
View Description Hide DescriptionThe states associated with the group‐subgroup are unambiguously representable as the stretched product states of just seven elementary multiplets. This result is used to derive a number of branching rules for the decomposition R _{7} → G _{2} in a simpler manner than hitherto available. An example of the decomposition of the spin representations of R _{7} is given.

On the Existence of Relations among the Eigenvalues of Difference Equations
View Description Hide DescriptionBy translating the eigenvalue problem of a difference equation to one of abstract operators, we give a method of finding relations among the eigenvalues of the difference equations.

Proof of an Existence Condition for Solutions of the Unitarity System of Equations
View Description Hide DescriptionA previously established set of conditions for the existence of solutions of a nonlinear system, expressing the unitarity of n × n symmetric matrices, is proven by means of the homotopy invariance theorem.

Geodesics and Classical Mechanics on Lie Groups
View Description Hide DescriptionHamilton‐Jacobi Theory on Lie groups is discussed, and an error in a previous paper is corrected. A method for ``analytically continuing'' geodesics from compact to noncompact Lie groups is presented.

Nonpolynomial Lagrangians with Isospin
View Description Hide DescriptionWe apply (Laplace, Fourier, …) transform methods to obtain compact representations of the perturbationS‐matrix elements for interaction Lagrangians which are general nonpolynomial functions of isospin multiplets. In order to illustrate the power and simplicity of the method for coping with the isospin complications, we have treated several examples in detail, and these include the commonly used parametrizations of the unitary chiral transformations.

A Class of Mean Field Models
View Description Hide DescriptionA model of isotropically interacting ν‐dimensional classical spins with an infinite range potential of the molecular field‐type is solved. The partition function is represented as the integral of e ^{−βHN } over an appropriate weight function, which, for given ν, is the Pearson random walkprobability distribution in ν dimensions. A molecular field‐type phase transition is obtained for all ν.

Exact and Simultaneous Measurements
View Description Hide DescriptionThe observables which can be determined through their states of exact measurement are characterized, and a criterion for the set of states in which two observables are both measured exactly is established so that the two observables will be compatible.

Spectrum of Local Hamiltonians in the Yukawa_{2} Field Theory
View Description Hide DescriptionIt is proved that the local Hamiltonians constructed for the two‐dimensional Yukawa model by Glimm and Jaffe have the same continuous spectrum as the free Hamiltonian. The proof depends on the construction of asymptotic creation and annihilation operators.

On the Mathematical Theory of Electromagnetic Radiation from Flanged Waveguides
View Description Hide DescriptionA mathematical technique particularly suited to describing the electromagnetic radiation from open‐ended waveguide structures is discussed. As an example of the utility of the method, the exact solutions, for both wave polarizations corresponding to a uniform line source embedded in a dielectric filled slot in a ground plane, are given. The edge condition is used to estimate and partially correct for the truncation error resulting from approximating the infinite system of linear equations by a finite one. The truncation corrected field expressions are shown to recover the proper local field behavior in the vicinity of the aperture perimeter. Some numerical results, for both aperture and radiation fields are given and in the latter case compared with the Kirchhoff approximation to the radiation field.

On Uniqueness of the Kerr‐Newman Black Holes
View Description Hide DescriptionIt is proven that the Kerr‐Newman space‐times with e ^{2} + a ^{2} < m ^{2} are the only electrovac black holesolutions of Einstein's equations which can be obtained by analytic variation of the space‐time geometry starting from the Schwarzschild solution.

Light‐Cone Singularities and Lorentz Poles
View Description Hide DescriptionThe causal Meyer‐Suura structure functions are projected into irreducible representations of the Lorentz group. A clarification of the connection between light‐cone singularities and Lorentz poles is obtained: We find that in general a light‐cone singularity of the type 1/(− x ^{2} + iεx _{0})^{α}, in the operator product of the hadronic electromagnetic current, is built up from a sequence of Lorentz poles at λ_{ n } = 1 + α − n whose residues are polynomials of order n in the virtual photon square mass.

Sliced Extensions, Irreducible Extensions, and Associated Graphs: An Analysis of Lie Algebra Extensions. I. General Theory
View Description Hide DescriptionThe Lie algebra extension problem is investigated and incorporated into a formalism which allows for a visualization of the algebraic structures involved. A finer analysis than the usual mathematical one is sometimes required for the physical applications. This brings about the consideration of sliced extensions, i.e., of extensions provided with sections. Some of these, the ω‐sliced extensions, are particularly interesting. They are directly connected with natural Levi decompositions of the Lie algebras obtained from extensions. Graphs are associated with ω‐sliced extensions. This is especially suitable for the study of irreducible extensions, which are the basic ones among the extensions. Their structure may be described in terms of graph theory. A subset is picked out from the set of all irreducible extensions of an arbitrary Lie algebra. Its elements, the primitive extensions, have the simplest extension structure and are characterized by the empty graph or by one‐vertex graphs.

Sliced Extensions, Irreducible Extensions, and Associated Graphs: An Analysis of Lie Algebra Extensions. II. Application to Euclidean, Poincaré, and Galilean Algebras
View Description Hide DescriptionThe results of a preceding paper on Lie algebra extensions and sliced extensions are applied to the Lie algebras of the Euclidean, resp. Poincaré and Galilean groups. The primitive extensions are analyzed in detail. A procedure for the construction of irreducible extensions is illustrated by some examples, using diagrams which picture the graphs of the extensions. It is proved that all extensions by are inessential.

Generalization of Euler Angles to N‐Dimensional Orthogonal Matrices
View Description Hide DescriptionAn algorithm is presented whereby an N‐dimensional orthogonal matrix can be represented in terms of ½N(N − 1) independent parameters The parameters have the character of angles, whose compact domains are defined in a manner such that there exists a one‐to‐one correspondence between the points in the parameter space and the group of orthogonal matrices. Explicit formulas are given which express all matrix elements in terms of the angles, and formulas are given which express the angles in terms of the matrix elements. Special choices of angles give block‐diagonal matrices. For three‐dimensional matrices, the parametrization is equivalent to that of Euler.

Dynamical Quantization
View Description Hide DescriptionAn intrinsic quantization procedure based on higher symmetries of classical dynamical systems and utilizing the techniques of van Hove and Souriau is proposed. The procedure is intrinsically Hamiltonian but not explicitly canonical in that the Heisenberg algebra plays no fundamental role. The proposed method is applied to the n‐dimensional harmonic oscillator and to the n‐dimensional hydrogen atom. This approach seems to provide the first intrinsic justification of the success of ordinary correspondential quantization for this last system.

Note on the Angular Momentum and Mass of Gravitational Geons
View Description Hide DescriptionIt is shown that (1) the angular momentum of a gravitational geon must be zero if it is axisymmetric and (2) the mass of a gravitational geon must be zero if it is stationary, i.e., if the space‐time possesses a Killing vector which is timelike at infinity. Here angular momentum and mass are defined in terms of the asymptotic form of the metric at large distances; they are physical quantities which can be experimentally measured by distant observers. Since the gravitational geons previously considered are highly dynamical on a small scale, our result on the vanishing mass of a stationary geon does not conflict with previous analyses showing that gravitational geons can have mass. Similarly, our results do not exclude the possibility of gravitational geons having nonvanishing angular momentum if they are not strictly axisymmetric.

The Einstein Equations of Evolution‐A Geometric Approach
View Description Hide DescriptionIn this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite‐dimensional geometry, we answer sharply the question: ``In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?'' By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body‐space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time‐dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the ``intrinsic shift vector field.'' The essence of our method is to extend the usual configuration space of Riemannian metrics to , where is the group of relativistic time translations and is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields . On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for as a degenerate metric on , however, the metric is quadratic in the velocity variables. The groups also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual ``constraint equations.'' A precise theorem is given for a remark of Misner that in an empty space‐time we must have We study the relationship between the evolution equations for the time‐dependent metric g_{t} and the Ricci flat condition of the reconstructed Lorentz metric g^{L} . Finally, we make some remarks about a possible ``superphase space'' for general relativity and how our treatment on is related to ordinary superspace and superphase space.