Volume 4, Issue 11, November 1963
Index of content:

A Lattice of Von Neumann Algebras Associated with the Quantum Theory of a Free Bose Field
View Description Hide DescriptionVon Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von Neumann algebra on another complex Hilbert space (the Fock space) is defined. This correspondence is proved to be an isomorphism between a certain complemented lattice of subspaces and that of the von Neumann algebras. This result has an application to the duality theorem in the theory of a free scalar field, which is to be discussed in a separate paper.
A necessary and sufficient condition on a subspace, in order that the corresponding von Neumann algebra is of type I, is obtained.

Method for Construction of Unitary Operators in Quantum Field Theory
View Description Hide DescriptionA method for the construction of explicit representations of unitary Hillbert‐space operators in the particle‐number representation is presented and illustrated by application to several examples.

Modified Langevin Equation for the Description of Brownian Motion
View Description Hide DescriptionA modified Langevin equation for the description of Brownian motion is shown to give results equivalent to those of the Langevin equation in some physical situations. The Smoluchowski equation for the probability density of the displacement of a Brownian particle is derived from the modified Langevin equation with the aid of assumptions weaker than those needed to derive the Smoluchowski equation from the ordinary Langevin equation. Sufficient conditions for the applicability of the modified Langevin equation to the calculation of configuration‐space averages are obtained. Then the modified Langevin equation is applied to three simple systems, and the results are compared with those of the Langevin equation.

Comparison of Two Generalizations of Maxwell's Equations Involving Creation of Charge
View Description Hide DescriptionA comparison is made of the Lyttleton‐Bondi, and the Watson theories for the electromagnetic fields produced when charge is created. It is shown that, except when dimensions of the order of the radius of the Universe are involved, the difference is negligible and that consequently the Watson theory, being mathematically more simple, is the better.

Existence and Uniqueness Theorems for the Neutron Transport Equation
View Description Hide DescriptionIn an attempt to understand the conditions under which the neutron transportequation has solutions, and the properties of those solutions, a number of existence and uniqueness theorems are proved. One finds that the properties of the solution are closely related to the boundedness of the source as well as to certain velocity‐space integrals of the scattering kernel. Both time‐dependent and time‐independent equations are considered as are also the time‐dependent and time‐independent adjoint equations. Although only a very few of all possible existence and uniqueness theorems for these equations are considered here, the work may serve as a guide to the treatment of similar problems.

Fundamental Properties of Perturbation‐Theoretical Integral Representations. II
View Description Hide DescriptionIn the first part of this paper, it is investigated, apart from the perturbation‐theoretical basis, under what conditions the perturbation‐theoretical integral representations can be derived, and two theorems are given concerning this problem. In the second part, the asymptotic behavior of the weight function in the integral representation is investigated in perturbation theory. It is proved that the weight function vanishes at infinity for an infinite sum over certain graphs which are much more general than the ladderlike graphs. This result gives the analyticity in the right half‐plane of complex angular momentum.

Singularities of Regge Trajectories and Asymptotes to Landau Curves
View Description Hide DescriptionA new class of singularities associated with the trajectories and residues of particular Regge poles is investigated. It is shown that the singularities are associated with properties of asymptotes to Landau curves. One of the singularities corresponds to the singularity of the Regge amplitude discovered recently by Islam, Landshoff, and Taylor. The only singularities affecting physical asymptotic behavior correspond to diagrams which have all three Mandelstam spectral functions.

High‐Energy Behavior in Perturbation Theory. II
View Description Hide DescriptionContributions to the asymptotic behavior of Feynman integrals are evaluated which correspond to pinches in the interior of the hypercontour of integration. It is shown that they give the Gribov‐Pomeranchuk phenomenon and Regge cuts. A set of diagrams is investigated which gives a Regge cut on the physical sheet.

Behavior of the Scattering Amplitude for Large Angular Momentum
View Description Hide DescriptionLanger's theory on the asymptotic behavior of the solutions of differential equations is applied to angular momentum, giving stronger results than were possible hitherto by Born approximation. It is shown that, for potentials V(r) analytic in the right‐hand r plane satisfying r ^{2} V(r) < ∞ at r = 0 and r = ∞, the phase shift has the asymptotic form (λ = l + ½),in the λ plane and for all complex k. Consequently, (a) all Regge trajectories are bounded for analytic potentials; there are no poles for λ → ∞ in the right‐half λ plane, (b) stronger limits can be given for the feasibility of the Watson‐Sommerfeld transformation. The pathological behavior of the cut off potentials (e.g., square‐well) is attributed to the nonanalyticity of the potential.

Asymptotic Behavior of Schrödinger Scattering Amplitudes
View Description Hide DescriptionThe behavior of the S‐wave scattering amplitude for large k is studied for potentials which vanish at infinity faster than any exponential, but are not cut off. The asymptotic behavior is very sensitive to the shape of the potential tail. If the potential decreases very rapidly, the growth of the Jost function resembles that with a cut‐off potential. If V(r) decreases only slightly more rapidly than an exponential, then f(k) exhibits a very rapid growth in the vicinity of the positive imaginary axis. In this case also the zeros of f(k) become very dense and are concentrated near the positive imaginary axis.

Coupled Schrödinger Equations and Statistical Boundary Conditions
View Description Hide DescriptionThe possibility of introducing distribution functions for the phase of a radial wavefunction at a boundary is studied for the case of elastic scattering. The properties of a simple class of distribution functions are discussed. The results are generalized to the solutions of an arbitrary number of coupled Schrödinger equations which satisfy time‐reversal invariance, provided there are no closed channels.

Canonical Variables for the Interacting Gravitational and Dirac Fields
View Description Hide DescriptionThe problem of reducing the Lagrangian for the interacting gravitational and Dirac fields to canonical form is discussed, using the vierbein formalism. The arbitrary gauge variables corresponding to local Lorentz transformations of the vierbein are removed by imposing Schwinger's ``time‐guage'' condition, and a further condition that the spatial part of the vierbein be symmetric. It is shown that in this guage the Lagrangian can be expressed in a canonical form involving essentially the same gravitational‐field variables as in the absence of matter, and that the generators of spatial translations and rotations have the expected form.

Cluster Development in an N‐Body Problem
View Description Hide DescriptionThe procedure for generating useful cluster development in problems dealing with the Jastrow wavefunction, as proposed by Wu and Feenberg, is discussed in detail. The existence of the expansion is proved to all orders; also a simple rule is given for computing the expansion coefficients. The result can be considered as a generalization of the Ursell‐Mayer formulas.

Strict Localization
View Description Hide DescriptionA complete characterization for a general quantum field theory is given of the strictly localized states introduced by J. Knight. It is shown that each such state can be generated from the vacuum by a partially isometric operator. Necessary and sufficient conditions are given for the superposition of such states to be also strictly localized. Finally, it is shown that there is a connection between the von Neumann type of the ring generated by the field operator in a finite region and the possibility of constructing strictly localized states.