Volume 7, Issue 3, March 1966
Index of content:

Error Bounds for Approximations to Expectation Values of Unbounded Operators
View Description Hide DescriptionA bound for (BΨ, Ψ) — (Bφ, φ), the error between the expectation value of a self‐adjoint operator B on an eigenfunction Ψ of a self‐adjoint Hamiltonian H and the value given by an approximating vector φ, is obtained when H is a radial Schödinger operator and B is multiplicative and unbounded. The analysis makes use of point and asymptotic estimates for Ψ.

On the Complex Structure of the Universe
View Description Hide DescriptionA complex space—time is constructed from physical axioms. Both gravitational and electromagnetic fields are approximations to parts of a geometric object defined on this complex space, while other parts may represent strong and weak interactions. The intersections of the singularities of the three independent geometric invariants are identified as elementary particles. This identification leads to geometric definitions of mass and momentum and suggests the geometric significance of internal quantum numbers.

On the L—S Basis of the Poincaré Group for the Case of Nonzero Rest Mass
View Description Hide DescriptionThe action of the generators of the Poincaré group on the basis functions spanning an irreducible representation [m ^{2}, s ^{2}] (m > 0) and labeled by the eigenvalues of the energy, orbital angular momentum and spin, and the projections on a fixed axis of the two latter are considered. The explicit canonical representation of the generators (1.1) are used. The formulas are derived first for the zero‐spin case (Sec. 2) and then for the general case (Sec. 3). It is shown that our L—S basis permits a much more compact and systematic derivation of the formulas than the basis involving total angular momentum and helicity, which is considered by Lomont and Moses. The separation of the orbital and spin parts allows us also to display explicitly the role played by the ``canonical'' definition of spin.

A New Method in the Theory of Potential Scattering
View Description Hide DescriptionA new method is given for investigating the position, number, and type of singularities the scattering amplitude may have as a function of complex momentum transfer. The advantage of this method over ones relying on the Watson transform is that one needs only to know the partial amplitudes for physical values of the angular momentum.

On the Coupling and Recoupling of Relativistic States
View Description Hide DescriptionSystems containing more than two relativistic particles are analyzed from the point of view of irreducible representations of the Poincaré group. Corresponding Clebsch—Gordan coefficients are calculated for three‐particle systems, and recoupling functions (which are the analog of the Racah coefficients) are defined.

Some Spatially Homogeneous Anisotropic Relativistic Cosmological Models
View Description Hide DescriptionSome solutions of the Einstein field equations for a dust source are given in explicit form. They are spatially homogeneous, irrotational, and anisotropic. They can be characterized as those spatially homogeneous expanding models that do not permit a simply transitive three‐parameter group of motions. The models are compared in detail with observations and with the Friedmann models. In a few instances slightly longer time scales are obtained with the present models than from the corresponding Friedmann models.

Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra
View Description Hide DescriptionFamilies of special functions, known from mathematical physics, are defined here by their recursion relations. The operators which raise and lower indices in these functions are considered as generators of a Lie algebra. The general element of the corresponding Lie group thus operates on the function in two ways: on the one hand it shifts the argument of the function; on the other hand it produces an infinite sum of functions (at the unchanged argument) with shifted indices. Equating the two results of the operation gives us ``addition theorems,'' hitherto derived by analytical methods. The present paper restricts itself to the study of 2‐ and 3‐parameter Lie groups.

Multipole Moments in Einstein's Gravitational Theory
View Description Hide DescriptionJanis and Newman have proposed definitions of multipole moments of axially symmetric gravitational fields in terms of the initial data on a characteristic surface. This paper extends their definitions to source distributions without symmetry by considering the electromagnetic and linear gravitational fields. The global definition of four‐momentum agrees, for sandwich waves, with the objects constructed by Mo/ller and Cornish in their attempts to provide a local definition of four‐momentum.

A General Theory of First‐Passage Distributions in Transport and Multiplicative Processes
View Description Hide DescriptionThe ``Milne problem,'' expressed in probabilistic terms, is solved for general transport and multiplicative processes. If a particle initially in a given state at a given position inside a surface τ is multiply scattered while traveling through a fixed medium, then given the scattering cross sections and, if required, the probability distribution for a change of state between collisions (e.f., by diffusion or ionization), the problem is to obtain the probability that the particle eventually effects a first passage through a specified position on the surface τ and in a specified state. In the case of a multiplicative process, the problem is, given in addition the rates of creation and annihilation of particles (considering the nature of the particle as a state variable), to obtain the probability that eventually n particles will emerge for the first time through specified positions on τ and in specified states (with n = 0, 1, 2, …). A general solution is given in the form of a convergent series whose terms are obtained by iteration; this solution is unique if and only if the probability θ_{∞} of an infinity of atomic events before a first passage (which is the limit of a certain nonincreasing sequence) is identically zero; in the multiplicative case θ_{∞} ≢ 0 may be taken to mean that the process is ``supercritical.'' The mathematical theory which leads to this solution is a generalization of the corresponding theory for time‐dependent Markov processes in which the time variable is replaced by a set of surfaces ordered by inclusion of their ``insides'' and is valid for Euclidean space of any number of dimensions. Applying it to the 4‐dimensional space of special relativity with ordered sets of spacelike surfaces, one obtains a Lorentz‐invariant formulation of the theory of physical Markov processes. A few examples are given.

Approximate Solution of Hill's Equation
View Description Hide DescriptionA modified WKB approximation, amenable to successive corrections, to the solution of the linear differential equation of second order having a periodic coefficient in normal form is presented. Considered as an application of the related equation method, it uses the free‐particle wave equation rather than, as in an alternative approach, Mathieu's equation. Particular attention is given the instances, where two simple turning points appear in the period and where there are no turning points. With respect to the one‐dimensional crystal, it is shown how the energy band structure can be gleaned directly from the given periodic potential.

Geometrization of a Complex Scalar Field. I. Algebra
View Description Hide DescriptionThe necessary and sufficient conditions that a Ricci tensor should algebraically represent the energy tensor of a complex scalar field are given.

Functional Differential Calculus of Operators. II
View Description Hide DescriptionThe calculus of operators developed in a previous paper is here carried through in momentum space. The differential quotient, i.e., the derivative of an operator with respect to a free‐field operator is expressed algebraically by means of generalized functions and certain commutators. No recourse is made to configuration space but the resultant calculus in p‐space is related to the previously developed one in x‐space by means of Fourier transforms. The calculus is presented for spins 0, ½, and 1. The difficulties which were encountered before in the calculus for charged vector fields are resolved by working with a field equation which incorporates the supplementary condition needed to eliminate the spin‐zero components.

Model of an Oscillating Cosmos which Rejuvenates during Contraction
View Description Hide DescriptionFor an oscillating cosmos we assume time‐symmetric initial conditions, invariant under t→ —t and t→ T + t, where t = 0, ±T, … are the times of maximal cosmos contraction. In particular we specify the same value U _{0} for the cosmos' ``internal energy'' U at all these times. The expectation value of an Heisenberg operator becomes now.Here A(0) is the projector into the space of the eigenvectors of U with eigenvalueU _{0}, and A(t) has the time dependence of an Heisenberg operator.
The time symmetry implies that the cosmos' oscillations are periodic and that expansions and contractions occur (except for local statistical fluctuations) time‐symmetrically to each other: If the entropy increases during expansion then it must decrease during contraction.
The general theory is applied to a highly simplified cosmos model in which no star condensation occurs. Assuming here that the initial state contains equal numbers of particles and antiparticles, the theory predicts a slightly inhomogeneous distribution of these particles such that during the cosmos' expansion not all particles annihilate, but a realistic density of particles and antiparticles (in different space regions) survives the expansion.

Euclidean Quantum Field Theory. I. Equations for a Scalar Model
View Description Hide DescriptionThe analytic continuations to imaginary time of the Green's functions of local quantum field theory define Euclidean Green's functions. Use of the proper‐time method allows to represent these functions as multiple Wiener integrals of functionals that obey infinite systems of coupled integral equations which are similar to, and for the particular model of a complex scalar field in quadrilinear self‐interaction considered here a limiting case of, systems studied in quantum statistical mechanics by Ginibre. As a consequence, the Euclidean Green's functions can for this model be obtained by a limiting process, with temperature and density going to infinity, from the reduced density matrices of a nonrelativistic Bose gas. Reduced functionals are defined and their equations determined as a prepartory step to renormalization in the super‐renormalizable cases of two and three dimensions.

On Variational Principles for Electromagnetic Theory
View Description Hide DescriptionNew variational principles for electromagnetictheory are established. A functional consisting of the field vectors is defined through the use of a convolution, and it is shown that the variation of this functional subject to appropriate constraints is completely equivalent to Maxwell's equations, Ohm's law, and the constitutive equations, together with appropriate boundary and initial conditions. The present formulation does not have the defects of the classical variational principle for electromagnetictheory since it does not require the introduction of scalar and vector potentials and a priori knowledge of the field vectors at the final stage. Two variational formulations for the electric and magnetic field vectors alone are also presented.

Phase Transition of a Two‐Dimensional Continuum—Ising Model
View Description Hide DescriptionWe decorate a plane two‐dimensional Ising model by placing on each bond of the lattice a continuum spin which is allowed to interact only with the Ising spins at the ends of the bonds. The continuum spins are chosen to be either Gaussian (i.e., normally distributed with zero mean and unit variance) or spherical (i.e., constrained to lie on the surface of a sphere) after Berlin and Kac, and by integrating out the continuum spins, the partition function of the decorated lattice is expressed in terms of the Onsager partition function of the plane two‐dimensional Ising model. The critical behavior of the model is as follows: For the Gaussian case, as for the plane Ising model, the specific heat has a logarithmic singularity at the critical pointT _{c} ^{ G } given byand as , the spontaneous magnetization goes to zero like t ^{1/8}. For the spherical case, the specific heat is continuous and has a cusp at the critical pointT _{c} ^{s} given by , with slope going to ± ∞ like as , and as t → 0^{+}, the spontaneous magnetization goes to zero like [t/ln t]^{1/8}.

Theorem on Gravitational Fields with Geodesic Rays
View Description Hide DescriptionIf one of the Ruse vectors of a field is assumed to be a geodesic having nonvanishing divergence θ, curl ω, and complex shear σ, the only vacuum metrics that exist are found to be of the cylindrical type, where the geodesic rays obey θ^{2} + ω^{2} = σσ̄.

Overlap Integrals between Atomic Orbitals
View Description Hide DescriptionGeneral formulas are developed for overlap integrals between Slater‐type atomic orbitals of arbitrary integer quantum numbers (nlm) and parameter values . The overlap integrals are expressed as finite trilinear forms of powers of ρ and certain auxiliary functions . The auxiliary functions are related to confluent hypergeometric functions and Jacobi polynomials, respectively, and stable recurrence procedures for their evaluation are given. The method is practical for use with an electronic computer and, even for high orbital quantum numbers, the loss in significant figures is found to be small for all parameter values. The overlap integral between two atomic orbitals (Anlm) and (Bn′l′m′) is shown to be proportional to .

Two‐Center Coulomb Integrals between Atomic Orbitals
View Description Hide DescriptionGeneral analytical formulas are developed for the two‐center Coulomb integrals between Slater‐type atomic orbitals of arbitrary integral quantum numbers (nlm). The Coulomb integral C is obtained by integrating the Poisson equation ΔC = —4πS, where S is the corresponding overlap integral. The Coulomb integrals are expressed as trilinear forms of powers and certain auxiliary functions G _{αβ} ^{γ}, H _{αβ} ^{γ}, J _{μν} ^{λ}, which are related to confluent hypergeometric functions and Jacobi polynomials respectively. Stable recurrence procedures for their evaluation are given, and use with an electronic computer program showed the loss in numerical accuracy to be small for all argument values as well as for high quantum numbers.

Solution of the Bloch—Nordsieck Model
View Description Hide DescriptionAn operator solution is constructed for the Bloch—Nordsieck model, in the formulation of Bogolubov and Shirkov. The following are some of the features of the solution. (a) The method of solution depends on the following assumption: The renormalized electron current is identical with the free‐field current. The resulting solution is consistent with this assumption. (b) The infraparticle structure of the electron is analogous to the infraparticle structure in Schroer's model. (c) The ordering of fields, i.e., the definition of their products, is considered in detail. A modification of Wick ordering (called F‐ordering in the text) appears particularly satisfactory. (d) The solution allows a heuristic discussion of the infrared representations of the electromagnetic field. These depend, apparently, on the chosen velocity of the electron.