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Volume 8, Issue 5, May 1967

New Approach to Unified Field Theory
View Description Hide DescriptionA new approach to a unified field theory combining Einstein's gravitational equations and the Maxwell equations is developed using a geometry in which GL(4, C) replaces GL(4, R) as the group of the principal bundle. The resulting equations are the same as Einstein's in the case of empty space but differ from those he proposed for the combined gravitational and electromagnetic fields. Charge is shown to appear only at singularities, and a symmetric solution corresponding to the classical Schwarzschild metric is presented.

Clebsch‐Gordan Series for Symmetrized Tensor Products
View Description Hide DescriptionAn explicit method for obtaining the Clebsch‐Gordan series for the symmetrized n‐fold direct products of a finite‐dimensional representation of a semisimple Lie group is discussed and illustrated. The method is based on the use of weight diagrams.

Complete Excitation Spectra
View Description Hide DescriptionA type of nonhomogeneous integral equation frequently encountered in interaction problems is solved by means of the Heaviside expansion theorem. The energy spectrum is obtained in the usual way from the solutions of the dispersionequation for the excitations, D(k, s) = 0. The momentum spectrum is obtained as the solutions of ∂D/∂k = 0, which is shown to be also a form of the Cauchy‐Riemann equations. It is proved that the existence of a solution of ∂D/∂k = 0 is a necessary condition for the existence of an analytic region on the momentum plane. It is also proved that the existence of a solution of dD/d k = 0 and ∂D/∂s = 0 is a necessary and sufficient condition for the existence of analytic regions in both variables, provided the group velocity is finite and continuous. As an example, two linearized self‐consistent equations with arbitrary coupling are solved.

Analytic Continuation of Laplace Transforms by Means of Asymptotic Series
View Description Hide DescriptionConditions under which a Laplace transform may be analytically continued, by means of an asymptotic expansion of F, outside the half plane of convergence of the Laplace transform integral are investigated. For t > k define for some fixed β with Re β < 1 and suppose that F is integrable on [0, k) for some k ≥ 0. First, it is shown that if R_{N} (t) = O{N!(σ/t)^{ N+1}} uniformly in N and t > k for some σ > 0, then the singular part of f at s = 0 can be determined in terms of a_{i} . If β is an integer, then in some neighborhood of s = 0 it is shown that,where g and h are analytic at s = 0 and !. If β is not an integer, in some neighborhood of s = 0 it is shown that , where , with g and h analytic at s = 0. Second, if the estimate on R_{N} (t) holds uniformly in N and in the complex t plane in the region for some λ > 0, then the analytic continuation of f can be determined in terms of the a_{i} . For any k′ > k and for arg s < π we have , where Γ is the incomplete gamma function and a(t) is the analytic continuation of !. If k = 0 in the hypotheses, then with a slight further restriction on F(t) one has . A generalization and application to a problem in nonrelativistic dispersion theory which includes a Coulomb potential are discussed.

Cluster Expansion of an Inverse Overlap Matrix for Solids
View Description Hide DescriptionIn this paper, a method is developed to compute an inverse overlap matrix based on a linked cluster expansion of a determinant. The inverse is expanded in terms of cluster integrals represented by diagrams and a recurrence relation for generation of all diagrams required is found. Thus the computation of an exact inverse overlap matrix is reduced to solving the recurrence equations self‐consistently. An approach for solving the equations is suggested and bounds for errors accompanying this procedure are calculated. The method is applied to the hydrogen lattice.

Simple Force Multipoles in the Theory of Deformable Surfaces
View Description Hide DescriptionThis paper is concerned with a nonlinear theory of simple force multipoles for a deformable surface, embedded in a Euclidean 3‐space; the surface is not necessarily elastic. The theory is developed with the use of basic thermodynamical principles, together with invariance conditions under superposed rigid body motions. For simplicity, the basic kinematical ingredients are restricted to be the (ordinary) monopolar velocity of the surface and suitable first‐ and second‐order gradients of the velocity. The theory of an elastic surface and other special cases of the general theory which bear on the foundations of the classical theory of shells are also discussed.

Bargmann‐Wigner Equations
View Description Hide DescriptionThe redundancy of the Bargmann‐Wigner equation for free particles of spin S greater than one‐half is analyzed. Only 2(2S + 1) equations involve an essential time derivative, as is well known. An additional ⅓(S + 3)(4S ^{2} − 1) is required to define all components of the representation. The remaining (S + 1)(4S ^{2} − 1) are time derivatives and divergence conditions; the latter occurring doubly.

Neutron Transport from a Point Source in Two Adjacent, Dissimilar, Semi‐Infinite Media
View Description Hide DescriptionA solution to the steady‐state, two‐dimensional Boltzmann equation is obtained for the flux due to a point source of neutrons located in a system of two adjacent half‐spaces having the same mean‐free path. Isotropic scattering and monoenergetic neutrons are assumed. The solution for an arbitrary source location is related to that for an interface source which is obtained by the Wiener‐Hopf technique. Asymptotic expansions for the flux far from the source, both along and away from the interface, are derived and compared with approximate theory.

Bounds for Certain Thermodynamic Averages
View Description Hide DescriptionUpper and lower bounds for thermodynamic averages of the form 〈{A, A†}〉 are presented.

Spinor Fields as Distortions of Space‐Time
View Description Hide DescriptionSpinor fields are introduced into Riemannian space‐time in a new way. This approach admits a simple geometrical interpretation of spinor fields. A linear connection for space‐time is derived which describes both gravitational and nongravitational forces. It is consistent with a straightforward generalization of the Dirac equation. This theory also entails a physical interpretation of inertial coordinates. The spin and current vectors of a spinor field are not in general orthogonal. This lack of orthogonality provides an absolute measure of one component of the gravitational field as seen in an inertial system.

Regularization and Peratization of Singular Potentials
View Description Hide DescriptionWe prove that the commonly used regularizations for singular potentials are successful. This means that one can investigate the peratization properties with confidence. A general argument for the success of peratization, as an approximation procedure, is presented. The class of failures of peratization is extended to a series of arbitrarily weakly singular perturbations on the inverse fourth potential. The two results present an unresolved contradiction; some resolutions are considered.

Radiation of a Point Charge Moving Uniformly over an Infinite Array of Conducting Half‐Planes
View Description Hide DescriptionThe excitation of an infinite array of parallel semi‐infinite metallic plates by a uniformly moving point charge is studied by the Wiener‐Hopf method. The problem is treated as a boundary‐value problem for the potential of the induced electromagnetic field, and is formulated in terms of a dual integral equation for the current density induced on the plates. The solution of the dual integral equation gives exact expressions for the induced current density and the induced field in the form of Fourier integrals. The Poynting vector is calculated, and the radiation shows that the array of plates behaves both like a diffraction grating and a series of parallel‐plate waveguides.

Absence of Ordering in Certain Classical Systems
View Description Hide DescriptionA classical inequality giving lower bounds for fluctuations about ordered states is derived. The inequality, analogous to a quantum result due to Bogoliubov, is established by a purely classical argument which makes explicit the nature of the surface boundary conditions required, a point which is rather obscure in the quantum derivations. As in the quantum case the inequality is useful in excluding certain kinds of phase transitions in one‐ and two‐dimensional systems. This is illustrated for several kinds of classical spin systems.

Three‐Dimensional Formulation of Gravitational Null Fields. II
View Description Hide DescriptionIn the previous paper of this series, it was shown that three types of gravitational null fields may be characterized on the analogy of the electromagnetic field. The first two types were discussed in the previous paper. This paper discusses the remaining third type (C) of gravitational null field. The necessary and sufficient condition that the gravitational field be of type C is obtained. The formalism is also extended to include nonempty gravitational fields. It is shown that the nonempty space‐time may also admit three types of gravitational null fields under certain circumstances. A typical case is discussed as an example.

Realizations of Lie Algebras in Classical Mechanics
View Description Hide DescriptionClassical Poisson bracket realizations of semisimple Lie algebras are considered. An attempt is made to determine the minimum number of canonical degrees of freedom needed to find a realization of a given Lie algebra. Under the restriction to the symmetric traceless tensor representations of the orthogonal groups, and the symmetric tensor representations of the unimodular unitary groups, it is shown that with n pairs of canonical variables one can find realizations of the Lie algebras of O(n + 2) and SU(n + 1), but no higher groups.

Properties of a Harmonic Crystal in a Stationary Nonequilibrium State
View Description Hide DescriptionThe stationary nonequilibrium Gibbsian ensemble representing a harmonic crystal in contact with several idealized heat reservoirs at different temperatures is shown to have a Gaussian Γ space distribution for the case where the stochastic interaction between the system and heat reservoirs may be represented by Fokker‐Planck‐type operators. The covariance matrix of this Gaussian is found explicitly for a linear chain with nearest‐neighbor forces in contact at its ends with heat reservoirs at temperatures T _{1} and T_{N}, N being the number of oscillators. We also find explicitly the covariance matrix, but not the distribution, for the case where the interaction between the system and the reservoirs is represented by very ``hard'' collisions. This matrix differs from that for the previous case only by a trivial factor. The heat flux in the stationary state is found, as expected, to be proportional to the temperature difference (T _{1} − T_{N} ) rather than to the temperature gradient (T _{1} − T_{N} )/N. The kinetic temperature of the jth oscillatorT(j) behaves, however, in an unexpected fashion. T(j) is essentially constant in the interior of the chain decreasing exponentially in the direction of the hotter reservoir rising only at the end oscillator in contact with that reservoir (with corresponding behavior at the other end of the chain). No explanation is offered for this paradoxical result.

Eigenfunction Expansions Associated with the Second‐Order Invariant Operator on Hyperboloids and Cones. III
View Description Hide DescriptionThe eigenfunction expansions associated with the second‐order invariant operator on hyperboloids and cones are derived. The global unitary irreducible representations of the SO _{0}(p, q) groups related to hyperboloids and cones are obtained. The decomposition of the quasi‐regular representations into the irreducible ones is given and the connection with the Mautner theorem and nuclear spectral theory is discussed.

Recursive Evaluation of Some Atomic Integrals
View Description Hide DescriptionA simple recursion scheme is set up for the higher derivatives of functions of the form and and applied to the rapid calculation of the Calais‐Löwdin two‐particle atomic integrals.

Exact Distributions of the Reduced‐Width Amplitude
View Description Hide DescriptionThe invariance hypothesis is used to derive the various multivariate distributions of the reduced‐width amplitude. Simple expressions are given for the multi‐level and multi‐channel distributions valid for all dimensions of the random orthogonal matrix.

Wigner Method in Quantum Statistical Mechanics
View Description Hide DescriptionThe Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.