Volume 7, Issue 2, February 1966
Index of content:

Structure of Space and the Formalism of Relativistic Quantum Theory. III
View Description Hide DescriptionThe elasticscattering of two scalar particles having equal masses is considered in the formalism of relativistic quantum theory over a Galois field GF(q). The scattering function σ determining the cross section is introduced. It is determined by the geometrical relations of Euclidicity, to be imposed on observable 4‐momenta in a finite geometry. Thus the requirement of Euclidicity of observable 4‐momenta can be considered as the counterpart, in a finite geometry, of the requirement of the analyticity of the invariant amplitude used in conventional S‐matrix theory for the determination of the cross section.

Structure of Space and the Formalism of Relativistic Quantum Theory. IV
View Description Hide DescriptionThe decay functions determining the lifetimes and the stability of particles in finite models of relativistic quantum theory are considered.

Applications of the Gel'fand‐Moshinsky Bases in Unitary Symmetry and its Breaking
View Description Hide DescriptionThe Gel'fand‐Moshinsky bases for the unitary irreducible representations of the SU _{3} group are applied to the unitary scheme for the classification of elementary particles and resonances. A unified method is given for the derivation of the matrix elements of the octet tensor operators in a way which makes transparent its application for other irreducible tensors. The results are given in a remarkably simple form. The systematic use of the basis provides a methodological alternative to the current tensorial methods.

Faddeev‐Type Equations for the Four‐Body Problem
View Description Hide DescriptionThe technique used by Faddeev to obtain connected equations for the nonrelativistic three‐body T matrix is generalized for four particles. It is shown that the four‐body equations are completely determined by the solutions of all the possible two‐body subsystems, as is the case in the three‐body problem. This approach can be extended to more complicated multiparticle systems.

Transfer‐Matrix Method for Gamma‐Ray and Neutron Penetration
View Description Hide DescriptionThe problem of radiationtransport is formulated in terms of a transfer matrix H which is a 2 × 2 matrix of operators. H is simply related to the more intuitive transmission and reflection operators T and R. An explicit expression for H is derived in slab geometry for radiation distributions that depend on the angle with the slab normal and on energy. H for a multilayer slab is the matrix product of the transfer matrices for the individual layers. A formal expression for H for a homogeneous slab of finite thickness is found in terms of the T and R appropriate to an infinitestimally thin slab. These in turn are related to the single‐scattering distribution and therefore can be computed from the microscopic cross section. For purposes of computation, finite matrix representations for the operators must be introduced corresponding to the finite vectors which approximate the distributions. Expansions of the distributions in the cosine of the angle and group representations in energy were chosen in the present work. Some numerical results are presented for gamma rays on aluminum. The extension to problems with internal sources and to nonplanar geometries is outlined.

Cartan Frames and the General Relativistic Dirac Equation
View Description Hide DescriptionThe Dirac equation for spin‐½ particles in curved space‐time is formulated using Cartan calculus. Unlike previous formulations, this method is easy to use because it expresses the Dirac equation in terms of well known objects like partial derivatives and special relativistic Dirac matrices. It allows a simple and direct treatment of neutrinos in homogeneous nonisotropic universes and in plane‐wave geometries. These solutions are compared and contrasted with the corresponding solutions containing electromagnetic radiation.

Expansion Theorem for Functions of Operators
View Description Hide DescriptionA method is given for expanding operator functions of q and p, where p = ℏ/i(∂/∂q), such that all q factors are to the left of the p factors. The method is applicable to the rearrangement of creation and annihilation operators.

Nonlinear Theory of Elastic Surfaces
View Description Hide DescriptionThe present paper develops a nonlinear theory for the deformation of an elasticsurface by assuming the existence of a strain energy function and postulating a principle of virtual work which governs its mechanical behavior. By considering the strain energy function to depend on the first‐ and second‐order deformation gradients, the field equations and the general constitutive relations are obtained. In addition to the conventional couple stresses, there are shown to exist energetically undetermined double stresses without moment.

The Generalization of Choh‐Uhlenbeck's Method in the Kinetic Theory of Dense Gases
View Description Hide DescriptionThe method proposed by Choh and Uhlenbeck to deal with kinetic phenomena in dense gases is generalized to all orders in the density. The set of integral equations for the functions defining the transport coefficients is derived. It is shown that the thermal conductivity and the shear viscosity are independent of the way in which the local temperature is introduced, namely, through the kinetic energy and through the total energy density. However, the bulk viscosity does depend on the particular definition of temperature. The relationship between the corresponding bulk viscosities is explicitly obtained.

The Stability of Many‐Particle Systems
View Description Hide DescriptionIt is shown that a quantal or classical system of N particles of distinct species α,β = 1, 2, … μ interacting through pair potentials φ_{αβ}(r) are stable, in the sense that the total energy is always bounded below by −NB, provided φ_{αβ}(r) exceeds some φ_{αβ} ^{(2)}(r) whose Fourier transform corresponds to a positive semidefinite μ × μ matrix for all p.
This result is applied to discuss ``charged'' systems and stability is proved for Coulomb interactions if the charges are somewhat smeared rather than concentrated at points. For a large class of potentials it is shown that classical instability implies quantum instability in the case of bosons and, in three or more dimensions, also of fermions. Quantum systems with Coulomb interactions (point charges) are discussed and it is shown in particular that their stability cannot depend on the ratios between the masses of the particles.

Axiomatic Foundations of Quantum Theories
View Description Hide DescriptionA formalism is developed on an axiomatic basis and is shown to contain as special cases classical mechanics, the usual quantum mechanics, as well as the quantum theory of systems with continuous superselection rules. The structure of the symmetries of a general physical theory is studied and a classification of observables is exhibited.

On the Reduction of Direct Products of the Irreducible Representations of SU(n)
View Description Hide DescriptionExplicit simple formulas are given for the reduction of some direct products of the irreducible representations of SU(n). All the interesting products from the point of view of the various symmetry models in particle physics are included as special cases of our results. In particular, all products of the totally symmetric and the regular representations of SU(n) are evaluated.

Monomer Pair Correlations
View Description Hide DescriptionIn this paper we evaluate the monomer pair correlation along the diagonals (p, p + 1) and (p + 1, p) of a square lattice otherwise packed with dimers. Using the perturbed Pfaffian technique the correlation can be expressed as a Toeplitz determinant i^{p} b _{ i−j+1}/2xgenerated by the function,where τ = x/y and x and y are the activities of x and y dimers. We will calculate the determinant exactly and prove that the correlation decays with increasing monomer separation as B/4r ^{½}; where B is simply related to the decay constant of the diagonal spin correlation at the critical point of a square ferromagnetic Ising lattice. The exact value is found to be B = 0.989487291.

On the Space‐Time Behavior of Schrödinger Wavefunctions
View Description Hide DescriptionFor the Schrödinger equation in R^{l} , with a potential V(x _{1} … x_{l} ) of the type considered by Kato, the following problem is solved: Given a monomial M(x _{1} … x_{l} ) of degree n in the coordinates, find sufficient conditions on the initial state u such that Me^{−iH t}u is continuous in t and increasing in norm not faster than t^{ n } as t → ∞. In the special case where V(x _{1} … x_{l} ) is a bounded C ^{∞}‐function with bounded derivatives, the result implies that (u, t) → e^{−iH t}u is a continuous mapping of S(R^{l} ) × R onto S(R^{l} ), S(R^{l} ) being the Schwartz space of rapidly decreasing functions in the usual topology.

New Approach to the Ising Problem
View Description Hide DescriptionThe partition function for the Ising model on a two‐dimensional rectangular lattice is cast into a form which closely resembles the vacuum expectation value of the S‐matrix in quantum field theory. The standard Onsager expression is obtained very simply in this formalism. It is further shown how the nonsoluble models can be expressed as field theories with quartic interactions, thereby resembling the standard many‐body fermion theories.

Numerical Solution of Non‐Fredholm (Singular) Integral Equations by Matrix Inversion
View Description Hide DescriptionIt is shown that the simple matrix‐inversion techniques often used in numerically solving linear integral equations with a Fredholm‐Schmidt (i.e., square‐integrable) kernel can also be employed for a wide class of non‐Fredholm (``singular'') equations. This class includes equations the kernel of which is the sum of a Fredholm‐Schmidt kernel and a kernel whose norm (in the operator sense) is less than one. In particular, the integral equations of the so‐called ``new strip approximation'' in particle dynamics belong to this class.

Metrical Lattice and the Problem of Electricity
View Description Hide DescriptionA reinterpretation of Einstein's tetrad geometry leads to new results if we abandon the over‐simplified picture of the vacuum as an almost empty Minkowskian manifold. The basic tetrad is identified with the four principal axes of the matter tensor which belongs to a strongly curved fourfold periodic Riemannian world. The macroscopic perturbation of the metrical lattice is investigated, corresponding to a mere rotation of the principal axes and assuming the quadratic action principle of general relativity. The perturbation Lagrangian yields the scalar E ^{2} − H ^{2}, (and thus the Maxwellianequations), although the basic manifold is strictly Riemannian, with a positive‐definite line element.

Methods of Quadrature for Euler Transform Integrals
View Description Hide DescriptionSeveral methods of treating Euler transform integrals exist. One such method follows from the expression of the Euler transform kernel as a bilinear series of independent solutions to the Jacobi equation valid for the integration variable in the real interval −1 to 1 and the transform variable outside. The transform function then is expressed as a series of solutions of the second kind to the Jacobi equation whose coefficients are the expansion coefficients of the function to be transformed in the complete set of Jacobi polynomials, provided the latter exist. Such a series is absolutely convergent for the transform variable not on the real interval cited above. Another method, due to MacRobert, permits quadrature of the Euler transform integral directly for certain integrands. Finally, the expansion of the Euler kernel in a bilinear series of Bessel functions and Neumann polynomials valid for the integration variable on the finite interval, 0 to a, is mentioned, and applied to several integrals. Examples of all three methods are given.

Orbits in a Magnetic Universe
View Description Hide DescriptionA cylindrically symmetric parallel bundle of magnetic lines of force, in equilibrium under their mutual gravitational attraction (``magnetic universe''), has recently received attention. While a Newtonian analysis suggests that the equilibrium is unstable, the complete general relativity analysis shows that the equilibrium is stable. This discrepancy may have to do with the unusually slow falloff of the gravitational field at large distances in this geometry. In order to understand the gravitational field of the static magnetic universe somewhat better, we have studied its timelike and lightlike geodesics (i.e., the orbits in it of electromagnetically neutral test particles with unit or zero rest mass). Since the density of magnetic flux‐and energy and stress and, therefore, ``gravitating mass''‐is approximately uniform in the vicinity of the axis, the motion of test particles there is like that in a Newtonian simple harmonic oscillator field. ``Vicinity'' here means within a small fraction ρ of the range radius ā = (6.96/B _{0}) × 10^{24} cm (B _{0} is the magnetic field on the axis measured in gauss). As is to be expected from the universality of the angular frequency ω_{0} in the harmonic oscillator field and the relation: orbital velocity ≅ ω_{0}ρ, no motion can get too far from the axis. Otherwise the physical orbital velocity would exceed the speed of light. It is in this way that the strength of the attractive field, though it does not remain strictly of the harmonic oscillator type as one proceeds outward, implies that there is a critical straddling radius ρ = 1/√3. Circular or circular helical light tracks occur only at the critical radius, and with B _{0} = 10^{5} G, the time required for light to circumnavigate the critical circle is about 200 years. The cylinder marked out by this radius plays a unique limiting role: All particles, whether of zero or nonzero mass, and no matter what their initial positions and velocities (except in the one singular subcase of light tracks parallel to the cylindrical axis), must have their orbits lying wholly or partially within the cylindrical region ρ < 1/√3; hence the use of the adjective ``straddling.'' Constants of motion which correspond closely to ζ‐component linear momentum, angular momentum, and energy in Newtonian mechanics are defined. Bounds are placed on these dynamical constants and on the apsidal radii by the requirement that the range of motion be real. Finally, the magnetic universe is complete in the sense that ``no news can enter or leave''‐all orbits are of infinite duration.

Correlation Functions for Eigenvalues of Real Quaternian Matrices
View Description Hide DescriptionThe eigenvalue density, the two‐, and the three‐point correlation functions for the ensemble of real quaternion matrices are calculated. The forms suggest a generalization for the n‐point correlation function. The probability that no eigenvalues lie inside a circle of radius r around the origin is also calculated for the ensemble of real quaternion matrices as well as for that of complex matrices. Upper and lower bounding functions for the last probability density are given.