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Volume 11, Issue 10, October 1970

Noncommutation Requirements and Field Dependence of Sources of Tensor Fields
View Description Hide DescriptionStarting with the noncommutation requirements among different components of a conserved source tensor for a massive spin‐2 field, we discuss the resulting noncommutation requirements between the source and the field variables, and the field dependence of the source that this necessitates.

Reduction of a Class of O(4, 2) Representations with Respect to SO(4, 1) and SO(3, 2)
View Description Hide DescriptionA complete classification of representations of SO(4, 2) with infinitesimal generators S_{AB} , characterized by the representation relation {S_{AB}, S^{A} _{C} } = −2ag_{BC} , and their extension by parity have been determined. The possible values of a are a = 1 − S ^{2}, These representations have then been reduced according to the two chains and The equivalence of these representations with the oscillator representations is established.

Approach to Equilibrium in a Simple Model
View Description Hide DescriptionThe time evolution of a class of generalized quantum Ising models (with various long‐range interactions, including Dyson's 1/r ^{α}) has been studied from the C*‐algebraic point of view. We establish that: (1) All 〈A〉_{ t } are weakly almost periodic in time; (2) there exists a unique averaging procedure over time; (3) the time evolution in the thermodynamical limit can be locally implemented by effective Hamiltonians in the algebra of quasilocal observables; (4) there exists a specific connection between the spectral properties of the time evolution of the initial state and the approach to equilibrium; (5) there are examples in which the time evolution is not G‐Abelian.

Classical Thermodynamics Simplified
View Description Hide DescriptionClassical thermodynamics is developed in a rigorous and quite general form. The approach is similar to Carathéodory's in that entropy and temperature are defined in terms of quantities which are more directly measurable, but Pfaffian forms and quasistatic processes do not appear. The mathematics used is elementary, apart from a small amount of symbolic logic and a very little topology.

Coulomb Green's Function in f‐Dimensional Space
View Description Hide DescriptionIt is shown that the f‐dimensional nonrelativistic Coulomb Green's function and the associated reduced Green's functions can be obtained by differentiation of the corresponding functions in the 1‐dimensional (f odd) or 2‐dimensional (f even) case. A new expansion of the 3‐dimensional coordinate‐space Coulomb Green's function and a new sum formula for a product of two Laguerre polynomials with different arguments are derived.

An Axiomatic Formulation of the Theory of Coinciding Simple Poles and Multiple Poles
View Description Hide DescriptionIn the Bethe‐Salpeter formalism, the scatteringGreen's function is known to have multiple poles synthesized out of coinciding simple poles. The present paper proposes an axiomatic approach to the problem of finding the residues of the multiple poles in terms of those of M coinciding simple poles. The latter residues are regarded as finite‐dimensional, mutually orthogonal projection operators on a reflexive Banach space and its dual. Then various properties of the residues of the multiple poles are derived without recourse to the original Bethe‐Salpeter equation, and especially it is shown mathematically that they can be decomposed into a direct sum of operators which commute with the Bethe‐Salpeter operator. The residues of multiple poles are explicitly determined in two particular cases, M = N + 1 and M = 2, where N denotes the highest order of the singularities (in a parameter) of the residues of the coinciding simple poles.

Lorentz Invariance in a Gravitational Field
View Description Hide DescriptionIn any theory of gravity in which free particles move along the geodesics of a 4‐dimensional metric tensor, a particular class of metrics can be defined which correspond to the fields of Newton's theory of gravity. In these Newtonian fields the metric coefficients which describe intrinsic properties of space and time are clearly separated from those that describe the gravitational field. This separation suggests an invariance in the gravitational field which is quite similar to the usual Lorentz invariance of electro‐magnetism. The infinitesimal form of the generalized Lorentz transformation is determined by the fact that the 3‐dimensional geometry remains Euclidean under the transformation. The finite form is determined so that the transformations form a group, and the group is found to be the usual Lorentz group. The transformation is then applied to fields that are not necessarily Newtonian.

Time Translations in the Algebraic Formulation of Statistical Mechanics
View Description Hide DescriptionWe present a new description of time translations in the C*‐algebraic formulation of statistical mechanics. This description is based on weaker assumptions than the hitherto accepted ones, due to Haag, Hugenholtz, and Winnink (HHW) [Commun. Math. Phys. 5, 215 (1967)]. It is shown that these weaker assumptions still lead to the principal results of HHW for Gibbs states and, further, that our assumptions, unlike those of HHW, are valid for the ideal Bose gas and strong‐coupling BCS models.

Integral Transformations in Momentum Space and Conformal Invariance
View Description Hide DescriptionA certain element Z of the identity component of the conformal group together with the Poincaré subgroup generate the whole conformal group. In order to prove the conformal invariance of an S‐matrix, only the invariance under Z has to be checked, once relativistic invariance has been established. The explicit form of Z for certain physically important representations of the different covering groups of the conformal group will be derived. The transformation Z turns out to be an integral transformation.

Relative Position Representation for a Relativistic Two‐Particle System
View Description Hide DescriptionTwo‐particle angular momentum states are constructed which are localized with respect to the magnitude of the relative position in the rest system and which have arbitrary 3‐momentum dependence. The associated relative position operator is constructed, and a quantum‐mechanical analog of the classical impact parameter is identified. Two‐particle angular momentum states are constructed, which are also localized with respect to the ``mean‐position'' of the 2‐particle system, and the associated ``mean‐position'' operator is seen to be a generalization of the 1‐particle Newton‐Wigner position operator.

Pure Thermodynamical Phases as Extremal KMS States
View Description Hide DescriptionWe compare the dynamical characterization of pure thermodynamical phases as extremal KMS states and their characterization as extremal time‐ or space‐invariant states. We find that, for a class of Weiss‐Ising models with periodic potentials, the extremal KMS states coincide exactly with the solutions of the self‐consistency equations familiar from molecular field methods. We show that the models considered are not η‐asymptotically Abelian in time. We conclude that the characterization of pure thermodynamical phases as extremal KMS states is the only correct one for these models. We pay special attention (in particular, in the decomposition of an arbitrary KMS state into its extremal KMS components) to the fact that the time evolution is not an automorphism of the C*‐algebra of the quasilocal observables.

Derivatives of Phase Shifts and Binding Energies by Use of Variational Principles
View Description Hide DescriptionThe Kohn‐Hulthén variational principle for the phase shifts, as well as the Rayleigh‐Ritz principle for the binding energies, are used to determine the derivatives of δ_{ l } = δ(V, E, l, m, ℏ) and E = E(V, l, m, ℏ) with respect to the listed parameters. A similar treatment utilizing Hamilton's variational principle leads to the corresponding classical results. The relation between the quantum mechanical and the classical expressions is examined. In particular, it is found that the quantum‐mechanical binding energy corresponds to a certain path average of the classical energy. Some applications of resulting formulas are briefly reviewed. This work is an extension of ideas originated by Fock and Demkov.

Labeling States and Constructing Matrix Representations of G _{2}
View Description Hide DescriptionA method is developed for labeling G _{2}‐internal states and for finding the matrices representing G _{2}‐generators. The simple Lie algebraG _{2} is embedded into A _{6}, whose representation spaces are labeled by Gel'fand patterns. For a given irreducible finite‐dimensional representation Ψ(G _{2}) of G _{2}, an optimal representation is chosen, and a lemma is formulated which enables us to select the subspaceR(Ψ) from the representation space R(Φ).

Rough‐Surface Scattering: Shadowing, Multiple Scatter, and Energy Conservation
View Description Hide DescriptionMultiple‐scatter and shadowing effects are included in an extended theory of high‐frequency scattering from a surface rough in one dimension. The single‐scatter probability of slopes relation, corrected for shadowing, is an immediate consequence for any stationary random process. The double‐scatter contribution (shadow corrected) is derived as well, and it provides a significant correction for surfaces with appreciable rms slope. The total power scattered by a perfectly reflecting rough surface is numerically evaluated as a test of energy conservation; the results show that the double‐scatter formulation is substantially more accurate than the conventional single‐scatter, unshadowed theory, particularly in the cases of large angles of incidence or very rough surfaces.

Boundary‐Value Problems of Linear‐Transport Theory—Green's Function Approach
View Description Hide DescriptionCase's technique utilizing Green's functions for dealing with boundary‐value problems of the neutron linear‐transport theory is exploited. We show that the Fourier coefficients of the Green's function over the Case spectrum are precisely the normal modes. In particular, if we assume that the scattering kernel is rotationally invariant (which indeed we do assume) and approximate it by a degenerate kernel consisting of spherical harmonics, the set of modes is deficient for problems lacking azimuthal symmetry. We also show that the expansion of the scattering kernel, in terms of spherical harmonics (or any set of orthogonal functions for that matter), permits the linear factorization of the Fourier coefficients of the Green's function in terms of the lowest element, with the proportionality functions consisting of complete orthogonal polynomials. As a consequence of this attribute of Fourier coefficients, the eigenfunctions (continuum and discrete) also factorize, which then permits decoupling of the appropriate singular integral equations. To illustrate our idea, we solve half‐space and slab problems. However, the basic procedure is kept sufficiently general so that the extension to problems involving other geometrics remains straightforward.

General Expansion of the Determinant of the Maxwell Dyadic
View Description Hide DescriptionElectromagnetic propagation, influenced by arbitrary tensor constitutive functions in an unbounded medium, is considered. The general expansion of the determinantal eigenvalueequation for the dispersion relations is obtained, exhibiting, for the first time, the functional dependence of the eigenvalueequation on the constitutive tensors.

Polynomial Algebras
View Description Hide DescriptionThe present work is concerned with what are called polynomialalgebras as an extension of the work of Ramakrishnan and his colleagues on the algebras of matrices satisfying conditions like L^{m} = I and L^{m} = L^{k} . Assuming L_{m} to be an m‐dimensional linear space, we generate a class of associative algebras called polynomialalgebras by requiring that every element L of L_{m} satisfy a polynomial equation L^{n} + P _{1} L ^{ n−1} + ⋯ + P_{n} = 0. We show that some very important algebras which physicists have found useful can be obtained by various restrictions on the polynomial. A few general properties of these algebras are established.

Symmetries of the Racah Coefficients
View Description Hide DescriptionA new symmetry of the Racah coefficients is derived using a property of a generalized hypergeometric function of unit argument. The symmetry is similar in appearance, though not derivation, to that given by Regge.

New Solutions of the Kinematic Dynamo Problem
View Description Hide DescriptionThe steady‐state kinematic dynamo problem in a homogeneous 3‐dimensional core is studied. The existence of a class of smooth solenoidal dynamos, satisfying a no‐slip condition on the core boundary, is proved using perturbation theory. The dynamos are of the form q = q ^{(1)} + q ^{(2)} + q ^{(3)}, where q ^{(1)} is spatially periodic on a sufficiently small scale of length, q ^{(2)} is zero except near the core boundary, and q ^{(3)} is an arbitrary sufficiently small motion. The term q ^{(1)} is also a spatially periodic dynamo in an appropriate sense for an infinite core. The last property allows a simple characterization of the bounded dynamos in terms of the admissible q ^{(1)}.

Multiplicities in the Classical Groups. I
View Description Hide DescriptionA unified treatment of all three multiplicities, valid for all classical (compact, connected, simple Lie) groups, is described. The general theory is given and then applied to the rank‐2 groups.