Volume 4, Issue 12, December 1963
Index of content:

On the Representations of the Semisimple Lie Groups. II
View Description Hide DescriptionThe explicit determination of the matrices of the generators of the unitary groups, SU_{n} , is carried out and discussed in two alternative treatments: (a) by purely algebraic infinitesimal methods, and (b) by Young‐pattern techniques employing the Schwinger‐Bargmann boson operator methods. The implication of this result for a tableau calculus is discussed and a determination of the [λ] × [1] Wigner coefficient for all SU_{n} is indicated.

A Class of Null Flat‐Space Coordinate Systems
View Description Hide DescriptionA new coordinate system, intrinsically attached to an arbitrary timelike world line, is investigated in flat‐space time. The Maxwell field tensor associated with the field of an arbitrarily moving charged particle assumes a particularly simple form in this, its intrinsic coordinate system. This reference frame is expected to be useful in General Relativity, in asymptotic studies of radiation, and equations of motion.

Interaction contra Classical Relativistic Hamiltonian Particle Mechanics
View Description Hide DescriptionThe relativistic Hamiltonian formalism is outlined and discussed for classical particles. The implications of the requirement that the coordinates of an event transform according to the Lorentz transformation law are discussed and expressed in a form, called the world‐line conditions, which may be considered in the relativistic Hamiltonian formalism. It is then shown that the world‐line conditions imply that there is no interaction in the relativistic Hamiltonian formalism; that is, the motion of any pair of particles described by the relativistic Hamiltonian formalism consists of straightline motion. In other words, if the events which compose the world lines of the particles transform according to the Lorentz transformation law, and the path of the particle is not a straight line, then this phenomena cannot be described in terms of a relativistic two‐particle Hamiltonian formalism. The experimental basis for a determination of the transformation properties of an event is considered, and the relationship of the experiment to the applicability of such a formalism is discussed.

The Remainder in Mayer's Fugacity Series
View Description Hide DescriptionUpper and lower bounds are obtained for the remainder after a finite number of terms of the expansions in powers of fugacity z for the pressure p, the s‐particle distribution functions, the density ρ, and the fugacity coefficient z/ρ, for a system of particles with two‐body interactions. The interaction potential must be either nonnegative or else have a hard core and decrease faster than r ^{−3} at large distances. The results hold for all positive z, and apply to lattice gases as well as to fluids.
For nonnegative potentials, the results imply that successive partial sums of each of these fugacity expansions provide alternate upper and lower bounds, valid even if the series diverges, on the physical quantity the expansion represents.

Integral Equations and Inequalities in the Theory of Fluids
View Description Hide DescriptionUsing the method of functional Taylor expansion developed previously, an extensive set of equations is obtained for the distribution functions and Ursell functions in a classical fluid. These include in a systematic way many previously derived relations, e.g., Mayer‐Montroll and Kirkwood‐Salsburg equations. By terminating the Taylor expansion after a finite number of terms and retaining the remainder, we also obtain inequalities for the distribution functions and thermodynamic parameters of the fluid. For the case of positive interparticle potentials, we recover the inequalities first found by Lieb. For nonpositive potentials, new inequalities (some also obtained by Penrose) are derived. These inequalities are applied to the case of a hard‐sphere fluid in three dimensions where they are compared with the results of machine computations and approximate theories. Different inequalities, not obtainable from the above considerations, and some properties of the fugacity expansions, are also derived.

Solution of Linear Integral Equations Using Padé Approximants
View Description Hide DescriptionIt is shown that the exact solution of a nonhomogeneous linear integral equation with a kernel K of rank n is given by forming the Padé approximant P(n, n) from the first 2n terms of the perturbation series solution. It follows that for a compact kernel K, the solution is lim_{ n→∞} P(n, n); this gives meaning to a large class of perturbation series when the perturbation is large. The possible extension of this result to wider classes of equations is discussed.

Remarks on the Type of Von Neumann Algebras of Local Observables in Quantum Field Theory
View Description Hide DescriptionThe von Neumann algebras of local observables associated with certain regions of space‐time are believed to be factors. We show that these algebras are not of finite type. The commutant of the tensor product of two semifinite von Neumann algebras is analyzed with the aid of this result. The factors in question have the vacuum state as separating and cyclic vector. It is shown that a factor of type I _{∞} with I _{∞} commutant, and a subfactor of type I _{∞} with I _{∞} relative commutant have a common separating and cyclic vector. This settles negatively some conjectures aimed at proving that these factors are not of type I. An argument of Araki's showing that the factors associated with certain regions are not of type I is presented in simplified form.

Perturbation Approach to the Electrostatic Problem for Irregularly Shaped Conductors
View Description Hide DescriptionA simple and straightforward perturbation method for treating the electrostatic problem of a charged, irregularly shaped conductor is presented. The perturbation solution is generated starting from the zero‐order solution for a charged sphere. The method consists of expanding the boundary condition in a Taylor series, which in effect transforms the boundary condition at the irregular boundary into a succession of boundary conditions to be satisfied at the surface of a sphere. The simplicity of the formalism consists further in applying, in a consistent manner, sufficient rather than necessary conditions on the successive correction potentials. First‐ and second‐order expressions for the potential, surface charge density, and capacitance of irregularly shaped conductors, are derived explicitly, and an elementary theorem for the first‐order capacitance is obtained. A perturbation expansion for the capacitance valid to all orders is presented. The application of the method is illustrated by calculating the capacitance of several irregularly shaped conductors. Possible generalizations to more complicated boundary‐value problems are indicated.

Theory of Transport Coefficients in Low‐Density Gases
View Description Hide DescriptionThe evaluation of the correlation‐function expressions for transport coefficients is discussed. It is shown that for the case of a low‐density nondegenerate monatomic gas with short‐range interactions, results are obtained which are in agreement with those usually found from the Boltzmann equation. The analysis makes use of a generalized master equation, and of the factorization theorem of Kac. While the main concern is with low‐density gases, the methods developed have a wider range of application.

Determinantal Solution of the Fredholm Equation with Green's Function Kernel
View Description Hide DescriptionThe Fredholm integral equation with a Green's function type of kernel has been transformed by Drukarev into an equivalent Volterra equation. It is now proven that the Neumann series solution of the Volterra equation yields the determinantal solution of the Fredholm equation. Thus, a Born approximation technique suffices to obtain the full Fredholm solution.

External‐Mass Singularity
View Description Hide DescriptionIt is shown by a concrete example that the scattering amplitude may have an external‐mass singularity, which does not correspond to a threshold for any energy variable. If the external mass exceeds the external‐mass singularity, the scattering amplitude (even for the forward scattering) is no longer a boundary value of an analytic function of energy variables in the usual sense. This would mean a deadlock of the S‐matrix theory based on the analyticity. In the Appendix an example of a threshold in the non‐Euclidean case is given.