Volume 7, Issue 6, June 1966
Index of content:

Unified Unitary Representation of the Poincaré Group for Particles of Zero and Positive Rest Mass
View Description Hide DescriptionAs is well known, Wigner's construction of the unitary representations in terms of little groups gives quite dissimilar forms for the cases m = 0 and m > 0, while the spinor representation which does give a unified description of the above two cases is not unitary. We point out that it is quite possible to give a unified unitary representation for both cases (Sec. 1). This is achieved simply by noting that, while the factorization of U(Λ) corresponding the choice of the little group of (1, 0) is not valid for m = 0, the explicit final expression for the Wigner rotation R _{w} and the corresponding infinitesimal generators remain perfectly well defined for m = 0, and continue to furnish a unitary (discrete spin) representation for this case, which is compatible with the restriction of helicity to a particular fixed value. Moreover, the representation thus obtained has a very simple and direct geometrical significance. The relation of our formulation with that of Wigner is studied (in Sec. 2) and the comparison with the spinor representations is given (Sec. 3). We rederive our representation, starting from a particular simple condition (4.1) (in Sec. 4) which holds for both cases (m = 0, m > 0). We then consider (Sec. 5) the application of our unified formulation to the reduction of direct products, involving particles of positive and zero rest mass, comparing the result with that of helicity coupling. In the Appendix we make certain remarks concerning the Hermiticity of the generators and the possibility of defining a position operator for m = 0. Comparison with Foldy's representation is given (in the Appendix and Sec. 3), explaining why his representation cannot be considered to be strictly unitary, though the relation with the unitary case is quite a simple one.

Generalized Discrete‐Continuum Radial Integrals with Coulomb Functions
View Description Hide DescriptionSimple closed‐form computational formulas involving only elementary functions are obtained for the hydrogenic, discrete‐continuum matrix elements <k, l′ r^{m}  n, l>. The results allow for different effective charge parameters Z and Z′ for the discrete and continuum functions. The central radial integral computed is sufficiently general to allow the evaluation of matrix elements of r^{m} exp(− r/ε) between the Coulomb continuum functions and any discrete function formed from r^{k} exp(− r/A). Results are also given for the free‐particle limit as Z′ → 0.

Nonlinear Theory of Elastic Directed Surfaces
View Description Hide DescriptionThe present paper develops a nonlinear theory for the deformation of an elastic directed surface by assuming the existence of a strain energy function and postulating a principle of virtual work which governs its mechanical behavior. The equations of equilibrium and the boundary conditions are shown to involve both the classical stress as well as the double stress. Constitutive equations are derived which give the stress and double stress as functions of a complete set of strain measures which describe the deformation of directed surfaces.

q‐Equivalent Particle Hamiltonians. I. The Classical One‐Dimensional Case
View Description Hide DescriptionThe classes of equivalent Lagrangians in one‐dimensional particle dynamics are found. These classes contain not only Lagrangians yielding the same equations of motion(Lagrangians differing by a total time derivative), but also those implying each other's equations of motion. The corresponding classes of Hamiltonians, all of which give the same orbits in configuration space, but in general different orbits in phase space, are also found. Some specific examples are presented.

On the Development of the Covariant Formulation of the Conservation Laws of General Relativity
View Description Hide DescriptionA method of obtaining Komar's covariant formulation of the conservation laws of general relativity directly from the variation of the scalar curvature density is presented. The procedure of obtaining this expression is free of the addition of arbitrary elements and only tensorial terms and operations are employed in the development.

Upper and Lower Bounds for Canonical Ensemble Averages
View Description Hide DescriptionUpper and lower bounds are obtained for canonical ensemble averages. The bounds are expressed entirely in terms of averages with respect to an arbitrary unperturbed canonical ensemble density operator. A weak form of the derived bounds is used to show that, for magnetic fields exceeding a given critical value, the magnetization of the antiferromagnetic linear chain approaches ferromagneticsaturation as the temperature approaches absolute zero.

Finite‐Dimensional Representations of Some Non‐Semisimple Lie Algebras
View Description Hide DescriptionWe study the finite‐dimensional representations of non‐semisimple Lie algebras. We give some general properties, and apply them to the case of the motion group and of the inhomogeneous Lorentz group.

Higher‐Order Poles in the S‐Matrix
View Description Hide DescriptionThe question of the possible existence of multiple poles in the S‐matrix is discussed. It is shown by means of a simple model that such poles are not generally incompatible with Lagrangianfield theory and can only be excluded in axiomatic formulations by the assumption of a unique correspondence between stable particles and the poles of the S‐matrix on the physical sheet.

Dynamics of Certain Spherical Charge Distributions
View Description Hide DescriptionTwo initial‐value problems are solved in the Lagrange representation. The initial configuration of the first is an isolated, negative, finite spherical gas. In the second a fixed, infinitely thin, positive, permeable, spherical shell is placed concentric and exterior to the negative gas of the first configuration. The charge of the total system is zero. The first system is totally soluble. Application to the special case of uniform initial density gives that the charge density at a point far removed from the initial sphere decays as the inverse cube of time, and is independent of radius. In the second configuration if a is the radius of the positive shell and b the initial radius of the negative gas, then for uniform initial distribution, all charge in the shell b(b/a)^{½} ≤ r ≤ b escapes. The remaining charge oscillates and is randomized by phase mixing.

The Most General Clebsch‐Gordan Coefficients of the Universal Covering Group of the Inhomogeneous Lorentz Group
View Description Hide DescriptionThe most general Clebsch‐Gordan coefficients to reduce the physically most important n‐fold product representations of the groups P̃ _{+}↑ and P̃ (Universal covering groups of the restricted and full inhomogemeous Lorentz groups) are derived. They are used to answer the question: What can be said about the S‐matrix if only Lorentz invariance is postulated?

QuantumMechanical Extension of the LebowitzPenrose Theorem on the Van Der Waals Theory
View Description Hide DescriptionRecently Lebowitz and Penrose gave a rigorous derivation of the van der WaalsMaxwell theory of the liquidvaportransition, and showed how the Maxwell equal arearule could be obtained from a proper statistical mechanical calculation. Their results are quite general—being valid in any number of dimensions and for a broad class of pair potentials—but they were proved only for classical mechanics. In the present work we extend the proof to quantum systems with any statistics—Boltzmann, Bose, or Fermi. One corollary of this extended theorem is a model of a Bose gas with a firstorder phase transition.

Stationary Dust‐Filled Cosmological Solution with Λ = 0 and without Closed Timelike Lines
View Description Hide DescriptionAn analytic and complete solution of Einstein's field equations without the Λ term is presented for a dust‐filled universe (p = 0). The solution is stationary and inhomogeneous and does not contain any closed time‐like lines. Also some of the properties of the solution are studied.

Nonhomogeneous Differential Equation with a Second‐Order Turning Point
View Description Hide DescriptionThe asymptotic behavior of solutions of a class of parameter‐dependent second‐order nonhomogeneous linear ordinary differential equations with a second‐order turning point is investigated. It is shown that, under certain conditions, particular solutions can be represented asymptotically by expansions involving certain special functions. Properties of these special functions are studied.

A Short Simple Evaluation of Expressions of the Debye‐Waller Form
View Description Hide DescriptionAverages like those encountered in the theory of the Debye‐Waller factor are evaluated in one sentence.

Derivation of the Generalized Boltzmann Equation in Quantum Statistical Mechanics
View Description Hide DescriptionThe hierarchy of the equations of motion for the reduced density matrices in quantum statistical mechanics is solved and the (cumulant) reduced density matrices at a time t are expressed in terms of those at an earlier time t _{0}. Diagrams are introduced to express the results. With the aid of the technique of partial summations, the general term in the kinetic equation for the one‐particle reduced density matrix or the generalized Boltzmann equation in quantum statistical mechanics is obtained. The equation is non‐Markovian. A method of reducing the equation to Markovian is sketched.

Theorem on the Clebsch‐Gordan Series in SU(n)
View Description Hide DescriptionA theorem on the Clebsch‐Gordan series conjectured by B. Vitale is proved using simple Young diagram techniques.

On Alternate Commutation Relations
View Description Hide DescriptionAn alternate quantization procedure for Bose fields is proposed. This procedure leads to a field algebra which is related to that of angular momentum rather than momentum. With this algebra is associated a maximum (n _{0}) to the number of particles allowed in a given momentum state. It is indicated by the investigation of models that physical observables converge for large n _{0} to those obtained using the canonical commutation relations.

An Axiomatic Approach to the Formalism of Quantum Mechanics. I.
View Description Hide DescriptionAn axiomatic formulation of a quantum mechanical formalism is given. The formulation is not in terms of objects associated with the Hilbert space, but in terms of a different kind of objects for which the name ``complex probability measures'' has been chosen. It is shown that the conventional Hilbert‐space formalism obeys the given axioms. A few consequences of the axioms are investigated, some of which are found useful in the second part of this work.

An Axiomatic Approach to the Formalism of Quantum Mechanics. II.
View Description Hide DescriptionAnother axiom has to be introduced to make the formalism given in Part I physically equivalent to the conventional Hilbert‐space formalism. Then it is shown that, given a certain fundamental set of observables, a B*‐algebra can be built into which the set O_{ b } of all bounded observables can be mapped injectively. The closure of the algebra generated by the image of O_{ b } into is itself. A Hilbert spaceH exists into which the set O_{0} of all pure physical states can be mapped injectively. The closure of the subset of H which is the image of this mapping is H itself. The algebra can be mapped in such a fashion into the C*‐algebra (H) of all bounded observables that these mappings provide, essentially, just a translation of the original formalism in a Hilbert‐space formalism.

The Dirac Bra and Ket Formalism
View Description Hide DescriptionThe place of the Dirac formalism in quantum theory is investigated using rigged Hilbert spaces. Emphasis is laid on the representation of observables by continuous linear operators and on the existence of sufficient eigenkets. Using the concept of labeled observables, a canonical procedure is given for constructing a rigged Hilbert space and the bra and ket spaces are constructed for nonrelativistic quantum systems of n interacting particles. Spectral theory is investigated in this framework and the results are compared with the Dirac formalism.