Volume 6, Issue 7, July 1965
Index of content:

Weight Factors for the Two‐Dimensional Ising Model
View Description Hide DescriptionA group G of local weights is constructed for the square, honeycomb, and triangular lattices which counts for any closed path in the lattice 1/2π times the change in the argument of the tangent vector (mod 2) and the number of enclosed units of area (mod 2). These weights are used to evaluate the partition function of the two‐dimensional Ising model with nearest‐neighbor interaction and with a particular, imaginary external magnetic field. For the square lattice, the method gives a result announced by Lee and Yang.

Conservation Laws for Free Fields
View Description Hide DescriptionThe recently discovered conservation laws for the ``zilch'' of the electromagnetic field (of which a simple derivation is presented) are examined in the context of a general discussion of bilinear conserved quantities in free‐field theories. It is shown that there is always an infinite set of these quantities, and a method of finding them all is presented and illustrated by applications to simple theories. The existence of these conserved quantities is shown to be a consequence of the fact that the momentum‐space density is constant in time.

Tensorial Description of Neutrinos
View Description Hide DescriptionIt is shown that the equation , where T _{μν} is a null antisymmetric tensor and T _{d} its dual, is necessary and sufficient that there be a two‐component Majorana neutrino field. That is, the neutrino field may be described solely in terms of tensorial quantities and operations, without the need for spinors.

Geometrization of a Massive Scalar Field
View Description Hide DescriptionA set of necessary and sufficient conditions involving only the metric tensor and the Einstein tensorG _{μν} are given which thereby guarantee that G _{μν} may represent the stress‐energy tensor of a massive real scalar field. One of the conditions is nonlocal, and is a demand that the energy tensor vanish at spatial infinity.

Bosons and Fermions
View Description Hide DescriptionIt is proven that one cannot construct boson creation and annihilation operators from a finite number of fermion operators. The proof follows from the isomorphism of the fermion algebra and the algebra of Dirac matrices.

Exact Eigenstates of the Pairing‐Force Hamiltonian. II
View Description Hide DescriptionThe restrictions on a previously reported class of exact eigenstates of the pairing‐force Hamiltonians are removed and it is indicated that all the eigenstates of this Hamiltonian can be included in this class. Explicit expressions are given for the expectation values of one‐ and two‐body operators in the exact, seniority‐zero eigenstates of this Hamiltonian. In particular, a simple expression for the occupation probabilities of the levels of the single‐particle potential is given. This expression may be easily evaluated for realistic nuclear systems.

Quantum Electrodynamics Without Indefinite Metric
View Description Hide DescriptionThe electromagnetic potentials A _{α}(x) are quantized in such a way that their space components are Hermitian and their time component anti‐Hermitian. On the other hand, the metric in Hilbert space and the Hamiltonian are positive definite. The above formalism, which leads to the usual commutator [A _{α}(x), A _{β}(y)] = ig _{αβ} D _{0}(x − y), is shown to be Lorentz‐covariant over the manifold of physical states. The latter contain neither longitudinal nor scalar photons (rather than an equal number of them, as in the usual theory). This definition is also Lorentz covariant. The S‐matrix is not unitary, but satisfies PS*PSP = P, where P is the projection operator over physical states. In other words, the S‐matrix is unitary over the subspace of physical states, this being sufficient for the interpretation of the theory.

Quantum Mechanical Time
View Description Hide DescriptionSome suggestions on the probabilistic structure of quantization are first obtained by studying local properties of random variables such as continuity and differentiability with respect to a topology in the sample space. This analysis is extended to analytic functions on the complex plane and conditions are formulated under which probability densities are expressed as sums of absolute squares of complex numbers. Physical restrictions are introduced through the Schrödinger equation for a single bound‐particle system. As a result, observations on a physical system become identified with the random selection of points in a topological measure space and the physical observables such as energy, position, and momentum, as well as time, are identified with measurablefunctions on appropriate spaces. Time, considered as a function, appears to be multiple‐valued with spacings between multiple values and a detailed functional structure that characterizes and is characterized by the physical system under observation and the physical observables being measured. Its detailed functional structure is related to the physically measurable probabilities of quantum theory and it is seen to serve in the capacity of a conditioning random variable in the computation of quantum mechanical expectations.

Algebraic Difficulties of Preserving Dynamical Relations When Forming Quantum‐Mechanical Operators
View Description Hide DescriptionProofs are presented showing impossibility of assigning differential operators (quantum observables) to classical mechanical observables in such a way as to preserve the usual bracket formalism. Difficulty is shown to arise even if we limit ourselves to preserving brackets between the Hamiltonian and a rather limited set of observables. Some other algebraic difficulties inherent in the operator assignment problem are also discussed.

Functional Expansion in Peratization Theory
View Description Hide DescriptionA functional peratization expansion is defined in the charged vector meson theory of weak interactions. The simplest approximation is carried through and shown to yield an allowed lepton‐leptonscattering amplitude, properly damped on the light cone. A basic deficiency in this procedure is the absence of any criteria for the estimation of the character and size of successive corrections.

Off‐Diagonal Long‐Range Order and Generalized Bose Condensation
View Description Hide DescriptionIt is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off‐diagonal long‐range order (ODLRO) in the single‐particle density matrix ρ_{1} provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single‐particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box‐enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box‐enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second‐quantizes with respect to Hartree‐Fock orbitals rather than free‐particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ_{1} is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ_{1}. The two‐particle density matrix ρ_{2} is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ_{2} and ODLRO of ρ_{2}, just as for ρ_{1}. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.

Schrödinger Basis for Spinor Representations of the Three‐Dimensional Rotation Group
View Description Hide DescriptionIt is shown that the double‐valued spherical harmonics provide a basis for the irreducible spinor representations of the three‐dimensional rotation group. Pauli's assertion to the contrary is shown to be false. Both infinitesimal and finite rotations are discussed in some detail. It is also shown that there remains a twofold degeneracy in the spherical harmonic Y_{im} when j and m are specified.

Asymptotic Reduced‐Width Amplitude Distributions
View Description Hide DescriptionThe method of moments is used to derive the reduced‐width amplitude distributions. The explicit dependence of the distribution function on the dimension N of the random orthogonal matrix for large values of N is obtained. It is shown that in the limit N → ∞, the distribution is the same as the one obtained using the explicit assumption of level independence.

Sufficient Conditions for an Attractive Potential to Possess Bound States. II
View Description Hide DescriptionA condition sufficient to secure the existence of a least one bound state for each angular momentuml ≤ L is given by the inequality,where q is an arbitrary constant and V(r) an everywhere‐attractive potential.

Variational Principle for Diffraction of Elastic Waves
View Description Hide DescriptionThe theory of diffraction of elastic waves is developed for the case of two homogeneous elastic media, welded together at an interface of arbitrary shape. The Green's function of the problem, defined to be the displacement at a point P _{1}, caused by a periodic force at a point P _{0}, is expressed in terms of the displacement and tension on the interface. A set of Fredholm integral equations for these interface functions is obtained. From them a variational principle is derived which gives the Green's function, minus the ``free'' Green's function, as the stationary value of a functional. It is analogous to the variational theorem of Levine and Schwinger for optical diffraction at an aperture in a screen. The variational equations of this functional are the above‐mentioned integral equations. Explicit expressions are obtained for the case of isotropic elastic solids and for liquids. A generalization to the case of pulsed waves is indicated.

Scattering by Singular Logarithmic Potential
View Description Hide DescriptionPotential of the form (gr ^{−4} ln^{2} r − g ^{½} r ^{−3})θ(r − r _{0}) is considered in connection with the applicability of peratization technique. The advantage of this potential is the fact that while it is dominated by a logarithmic part near the origin, the exact solution of the zero‐energy and s‐wave Schrödinger equation is obtained in a closed form. We show that the peratization technique gives the correct answers.

Evaluation of Feynman's Functional Integrals
View Description Hide DescriptionSeveral relationships for Feynman's functional integrals are derived. From these relationships, we construct two different schemes for approximating Feynman's functional integral. The methods of approximation are expected to converge sufficiently rapidly in many cases so that only the lowest orders of the approximation are required to give reliable answers.

Time‐Dependent One‐Speed Albedo Problem for a Semi‐Infinite Medium
View Description Hide DescriptionA Laplace transformation technique is used to determine the neutron distribution in a semi‐infinite medium which has been irradiated by a neutron pulse. The result is given in terms of known solutions of Milne's problem and of the steady‐state albedo problem, which in turn are expressed by aid of Case's X‐function. Simple asymptotic approximations, valid for t ≫ 1, are deduced from the exact result.

Asymptotic Expansions of Solutions of Differential Equations
View Description Hide DescriptionA generalization of Ford's method, concerning the asymptotic expansions of solutions of differential equations with polynomial coefficients and with three or more regular singular points and one irregular at infinity, is presented. The analysis is subsequently extended to the special case of integral values for the difference of exponents of the differential equation, thus providing the complete asymptotic expansion of the second, logarithmic solution of the equation. Explicit formulas for the evaluation of the constant coefficients of the expansions are given; each coefficient is expressed in terms of a single solution of the adjoint difference equation associated with the original differential equation.Differential equations possessing singularities in excess of the hypergeometric equation (3 regular) or its variations, appear as separated solutions of the wave equation in certain stratified media whose index of refraction is a continuous function of position.

On Polynomial Systems in a Banach Ring
View Description Hide DescriptionWe define and discuss equations on Banach rings(algebras) which are of polynomial form. We prove a local uniqueness theorem for the homogeneous case, and an existence and local uniqueness theorem for the nonhomogeneous case. In order to apply these results to the equations of Lagrangianquantum field theory we find it necessary to extend the concept of a ring to that of an n‐ring. The resulting theory is applied to a simple modelequation arising in quantum field theory.