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Volume 10, Issue 1, January 1969

Curvature and the Petrov Canonical Forms
View Description Hide DescriptionThe Petrov classification for the curvature tensor of an Einstein space M ^{4} is related to the critical‐point theory of the sectional‐curvature function σ, regarded as a function on the manifold of nondegenerate tangent 2‐planes at each point of the space. It is shown that the Petrov type is determined by the number of critical points. Furthermore, all the invariants in the canonical form can be computed from a knowledge of the critical value and the Hessian quadratic form of σ at any single critical point.

Translational Invariance Properties of a Finite One‐Dimensional Hard‐Core Fluid Using the Grand Canonical Ensemble
View Description Hide DescriptionTranslational invariance properties of the single‐particle distribution functionD _{1}(x, L, z) of the grand canonical ensemble are investigated for a one‐dimensional hard‐core fluid. For a fluid of finite length L it is shown that D _{1}(x, L, z) is nowhere constant. It is shown that, in the thermodynamic limit and for x far from either wall, D _{1}(x, L, z) is a constant equal to the grand canonical density ρ.

Instability of Transverse Waves in a Relativistic Plasma
View Description Hide DescriptionLinearized equations are set up to describe disturbances in an infinite, spatially uniform, relativistic plasma without an ambient magnetic field. It is shown that, as well as the usual electrostatic waves, there also exists a class of electromagnetic waves. The two sets of waves are coupled in general, but can still be classified as mainly longitudinal or mainly transverse. Under the assumption that the system is stable against the longitudinal disturbances it is shown that the relativistic plasma will be unstable to the transverse waves unless it is virtually isotropic.

Unitary Representations of SL(2, C) in an E(2) Basis
View Description Hide DescriptionStarting from the functional representation of Gel'fand and Naimark, the unitary irreducible representations of SL(2, C) are described in a basis of the subgroup , where is the subgroup of all 2 × 2 matrices of the form , . Physically, this is the subgroup into which SL(2, C) degenerates at infinite momentum and may be thought of as the 2‐dimensional Euclidean group together with its dilations. Advantages to using the basis are: (1) It is convenient to calculate form factors; (2) the generators of are represented either multiplicatively or by first‐order differential operators and are independent of the values of the SL(2, C) Casimir operators; (3) the principal and supplementary series of SL(2, C) are treated on the same footing and, in particular, have the same inner product; and (4) the transformation coefficients to the usual angular‐momentum basis are related to Bessel functions. The is used to compute explicitly the finite matrix elements of an arbitrary Lorentz transformation and to investigate the structure of vector operators in unitary representation of SL(2, C).

Configuration‐Space Approach to the Four‐Particle Problem
View Description Hide DescriptionThe configuration‐space approach to the three‐particle problem is generalized to the case of four particles. Special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles. The construction of a suitable orthogonal system is discussed. Some of these functions are given explicitly. It is pointed out that the use of this orthogonal system leads to a considerable simplification for a large number of four‐particle problems, namely, the approximate reduction of the Schrödinger equation to a finite system of coupled differential equations for functions that depend on one variable only.

Solutions of the Zero‐Rest‐Mass Equations
View Description Hide DescriptionBy means of contour integrals involving arbitrary analytic functions, general solutions of the zero‐rest‐mass field equations in flat space‐time can be generated for each spin. If the contour surrounds only a simple (respectively, low‐order) pole of the function, the resulting field is null (respectively, algebraically special).

Many‐Neighbored Ising Chain
View Description Hide DescriptionUsing a method suggested by Montroll, we extend the well‐known matrix formulation of the nearest‐neighbor one‐dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic fieldB. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)^{ n }. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific‐heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.

Comments on the Classical Theory of Magnetic Monopoles
View Description Hide DescriptionThe classical theory of electromagnetism including magnetic monopoles is formulated in terms of harmonic functions. The fact that there is no consistent action‐integral formulation of the field that yields both particle and field equations for both electric and magnetic charges is discussed in detail. It is seen that a consistent formulation can be developed through an action integral, but, in such a development, a monopole does not have what has been considered to be an appropriate interaction with either an electric charge or another monopole.

Note on Two Binomial Coefficient Identities of Rosenbaum
View Description Hide DescriptionThis paper gives rapid proofs of two binomial coefficient identities found by Rosenbaum [J. Math. Phys. 8, 1977 (1967)] who obtained the identities from rather involved considerations of commutation relations. The present proofs make use of the Vandermonde convolution, or addition, theorem and a well‐known fact that the kth difference of a polynomial of degree k − 1 is zero. In a sense the two special cases are not essentially new.

Partially Alternate Derivation of a Result of Nelson
View Description Hide DescriptionThe result of Nelson that the total Hamiltonian is semibounded for a self‐interacting Boson field in two dimensions in a periodic box is derived by an alternate method. It is more elementary in so far as functional integration is not used.

Dirac Formalism and Symmetry Problems in Quantum Mechanics. I. General Dirac Formalism
View Description Hide DescriptionDirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out. These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, i.e., a triplet where is a Hilbert space, Φ a dense subspace of provided with a new (finer) topology, Φ′ the dual of Φ. It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle. The temporal evolution of the system and the wave equation are considered. Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all possible experiments (apparatus) to which it can be subjected; this provides a direct connection with Feynman's formulation of quantum mechanics.

Killing Horizons and Orthogonally Transitive Groups in Space‐Time
View Description Hide DescriptionSome concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n‐dimensional pseudo‐Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n ‐ p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n ‐ 2)‐parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space‐time (n = 4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source‐free Maxwell‐Einsteinequations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure‐vaccuum case).

High‐Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission
View Description Hide DescriptionThis is Paper I of a series on high‐frequency scattering of a scalar plane wave by a transparent sphere (square potential well or barrier). It is assumed that , where k is the wave‐number, a is the radius of the sphere, and N is the refractive index. By applying the modified Watson transformation, previously employed for an impenetrable sphere, the asymptotic behavior of the exact scattering amplitude in any direction is obtained, including several angular regions not treated before. The distribution of Regge poles is determined and their physical interpretation is given. The results are helpful in explaining the reason for the difference in the analytic properties of scattering amplitudes for cutoff potentials and potentials with tails. Following Debye, the scattering amplitude is expanded in a series, corresponding to a description in terms of multiple internal reflections. In Paper I, the first term of the Debye expansion, associated with direct reflection from the surface, and the second term, associated with direct transmission (without any internal reflection), are treated, both for N > 1 and for N < 1. The asymptotic expansions are carried out up to (not including) correction terms of order (ka)^{−2}. For N > 1, the behavior of the first term is similar to that found for an impenetrable sphere, with a forward diffraction peak, a lit (geometrical reflection) region, and a transition region where the amplitude is reduced to generalized Fock functions. For N < 1, there is an additional shadow boundary, associated with total reflection, and a new type of surface waves is found. They are related to Schmidt head waves, but their sense of propagation disagrees with the geometrical theory of diffraction. The physical interpretation of this result is given. The second term of the Debye expansion again gives rise to a lit region, a shadow region, and a Fock‐type transition region, both for N > 1 and for N < 1. In the former case, surface waves make shortcuts across the sphere, by critical refraction. In the latter one, they excite new surface waves by internal diffraction.

High‐Frequency Scattering by a Transparent Sphere. II. Theory of the Rainbow and the Glory
View Description Hide DescriptionThe treatment, initiated in Paper I [J. Math. Phys. 10, 82 (1969)], of the high‐frequency scattering of a scalar plane wave by a transparent sphere is continued. The main results here are an improved theory of the rainbow and a theory of the glory. The modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections. Only the range , where N is the refractive index, is considered. In the geometrical‐optic approximation, this term is associated with rays transmitted after one internal reflection, and there are three angular regions, corresponding to one ray, two rays, or no ray (shadow) passing through each direction. Together with transition regions, this leads to six different angular domains. In the 1‐ray and 2‐ray regions, geometrical‐optic terms are dominant. Correction terms corresponding to the 2nd‐order WKB approximation are also evaluated. In the 0‐ray region, the amplitude is dominated by complex rays and surface waves. The 1‐ray/2‐ray transition is a Fock‐type region. The rainbow appears in the 2‐ray/0‐ray transition region. The extension of the method of steepest descents due to Chester, Friedman and Ursell is applied. The result is a uniform asymptotic expansion for the scattering amplitude. It reduces to Airy's theory in the lowest‐order approximation, but its domain of validity is considerably greater, both with regard to size parameter and to angles. The glory is an example of strong ``Regge‐pole dominance'' of the near‐backward scattering amplitude. Van de Hulst's conjecture that surface waves are responsible for the glory is confirmed. However, besides surface waves taking two shortcuts through the sphere, higher‐order terms in the Debye expansion must also be taken into account. By considering also the effect of higher‐order surface‐wave contributions, all the features observed in the glory (apart from the polarization) are explained. Resonance effects associated with nearly‐closed paths of diffracted rays lead to large, rapid, quasiperiodic intensity fluctuations. The same effects are responsible for the ripple in the total cross‐section. Similar fluctuations appear in any direction, but their amplitude increases with the scattering angle, becoming a maximum near the backward direction, where they are dominant. They can also be interpreted as a collective effect due to many nearly‐resonant partial waves in the edge domain. The dominant surface‐wave contributions can also be summed to all orders for N < 1, leading to a renormalization of the propagation constants of surface waves.

Problem of Two Spin Deviations in a Linear Chain with Next‐Nearest‐Neighbor Interactions
View Description Hide DescriptionThe problem of two spin deviations from the fully aligned state is studied in a linear chain for the Hamiltonian [J(i,j) > 0]:,where (nn) and (nnn) mean nearest‐ and next‐nearest‐neighbor interactions, respectively. The behavior of the bound state, which is found to exist for α ≥ 0 only, is discussed.

Some Asymptotic Behavior of Stieltjes Transforms
View Description Hide DescriptionWe consider an integrable function g(t) which behaves as when t tends to infinity is a finite number) for , and show that its Stieltjes transform has the same behavior when z approaches infinity and provided that z is in the sector . (Theorem 1). In addition, we study the cases of α equal to zero, one, and larger than one (Theorems 2–4). Our results contain those of L. Lanz and G. M. Prosperi [Nuovo Cimento 33, 201 (1964)] and those of W. S. Woolcock [J. Math. Phys. 8, 1270 (1967)]. They are proved in a direct manner, using a theorem of D. V. Widder [The Laplace Transform, (Princeton University Press, Princeton, N.J., 1959), fifth printing, p. 329] and the regularity of the integral transforms that arise.

Bethe‐Salpeter Equation
View Description Hide DescriptionWe treat the Bethe‐Salpeter equation as a problem in singular integral equations. As such, it has three outstanding features: its algebraic structure, the fixed propagator singularities in the direct channel, and the possible singularities in the potential, which are usually moving singularities. We exploit the algebraic structure in order to give insight into the possible correctness classes for the equation. We give explicit prescriptions for the removal of fixed singularities in a wide class of equations. We show under what circumstances these prescriptions can be adapted to maintain such desirable features as symmetry of the kernel. Moving singularities arise in physically realistic kernels; they are the crossed‐channel singularities. The basic mathematics of such singularities is well known and is related to the Riemann‐Hilbert problem, but this is useless in off‐shell methods because it cannot cope with the integration over the space parts of 4‐momenta. Instead, we adopt a method (proposed by one of us elsewhere) based on analyticity in energy variables. The resulting formalism is too complicated to be applied in full generality. We therefore consider the example of the single‐particle exchange potential in detail, and show how the moving singularities can be eliminated, exhibiting the resulting equations explicitly in a form to which our theory of fixed singularities can immediately be applied. All our arguments are exact.

Kinetic Description of an Inhomogeneous Plasma
View Description Hide DescriptionThe general description of a strongly inhomogeneous one‐component plasma in the so‐called ``ring'' approximation is derived. Using the general theory of inhomogeneous systems, the closed system of two equations in one‐particle phase space is obtained. The additional equation for some function, which appears in the collisions term, has the form of a Vlasov equation linearized around the inhomogeneous one‐particle distribution function. The meaning of the parameters which appear in this equation is discussed. This equation is solved in the hydrodynamic approximation. The collision operator in the Markoffian limit reduces to the well‐known form. The velocity distribution function for the inhomogeneous state is discussed and some additional terms to the usual Balescu‐Guernsey‐Lenard equation, in the case of no square‐integrable inhomogeneity factors, are obtained. The influence of initial correlation is discussed.

Combinatorial Structure of State Vectors in U_{n} . I. Hook Patterns for Maximal and Semimaximal States in U_{n}
View Description Hide DescriptionIt is shown that, in the boson‐operator realization, the state vectors of the unitary groups U_{n} —in the canonical chain —can be obtained ab initio by a combinatorial probabilistic method. From the Weyl branching law, a general state vector in U_{n} is uniquely specified in the canonical chain; the algebraic determination of such a general state vector is in principle known (Cartan‐Main theorem) from the state vector of highest weight; the explicit procedure is a generalization of the SU(2) lowering‐operator technique. The present combinatorial method gives the normalization of these state vectors in terms of a new generalization of the combinatorial entity, the Nakayama hook, which generalization arises ab initio from a probabilistic argument in a natural way in the lowering procedure. It is the advantage of our general hook concept that it recasts those known algebraic results into a most economical algorithm which clarifies the structure of the boson‐operator realization of the U_{n} representations.