Volume 8, Issue 3, March 1967
Index of content:

Singular Potentials and the Z = 0 Condition
View Description Hide DescriptionWe consider an interaction Lagrangian consisting of renormalizable and nonrenormalizable terms. After a brief discussion of the vertex function and the boundary conditions determining the renormalization constant corresponding to a pseudoscalar bound state, we derive an explicit expression for the nonrelativistic S‐matrix for scattering by the potential 1/r ^{4}. These results are then used to evaluate Z. The explicit expressions obtained contain a strong cut‐off dependence which cannot be factorized out. However, a specific value of the cut‐off reduces the equation to a well‐known eigenvalue problem, so that in this case a discrete spectrum of bound states may be obtained.

Non‐Usual Topologies on Space‐Time and High‐Energy Scattering
View Description Hide DescriptionMotivated by recently observed deviations from quantum electrodynamical theory, we study the possibility that our notions of space‐time may need revision at small distances. In this work, we wish to call attention to certain techniques which are available for studying different space‐time structures within the framework of topology. Our main effort is in the consideration of a non‐usual topology on space‐time in which is embedded an elementary length. By working separately in each n‐particle subspace, the embedding is done in an inhomogeneous Lorentz invariant way, and we avoid any lattice structure in space‐time. Particles in this topology are in general extended structures, and we find the surprising feature that, at high energies, the topology enhances backward and large‐angle scattering. From these preliminary investigations, we are not as yet able to make more than qualitative comparison with experiment. Along the way, we have the opportunity to remark on ways of embedding an intrinsic breakdown of certain invariances (e.g., parity) in the topology of space‐time.

Modification of the Ehrenfest Model in Statistical Mechanics
View Description Hide DescriptionThe Ehrenfest model has been used to explain the ``irreversibility'' of thermodynamics and statistical mechanics. The modification described in this paper allows transitions to occur in both directions between the two ``boxes'' at each step of the model procedure. The equilibrium probability distribution is given in the form of a finite product, or in an iterated form particularly suitable for machine calculation. The analysis is illustrated by a simple model of an ionization‐recombination process.

Thermodynamically Equivalent Hamiltonian Method in Nonequilibrium Statistical Mechanics
View Description Hide DescriptionThe equilibrium statistical mechanics of the Bardeen‐Cooper‐Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbationV which is spatially localized, but not necessarily small.

Local Weights Which Determine Area, and the Ising Model
View Description Hide DescriptionA group G of local weights is constructed which assigns to closed paths in the square lattice the enclosed area and the number of turns of the tangent vector (mod 2) to the path. Special cases of this group have been used previously in explicit evaluations of the partition function for the Ising model in 2‐dimensions. Properties of G are examined to cast light on the combinatorial approach to the Ising problem developed by Kac and Ward, Feynman, and Sherman. It is shown that their method breaks down in the general case.

New Characterization of the Ray Representations of the Galilei Group
View Description Hide DescriptionAn elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy‐Leblond,.

Time‐Dependent Green's Function for Electromagnetic Radiation in a Conducting Moving Medium: Nonrelativistic Approximation
View Description Hide DescriptionThe electromagnetic radiation emanating from a source immersed in a linear, homogeneous, conducting medium moving with a uniform velocity v with respect to the rest frame of the source distribution is investigated. It is shown that in the nonrelativistic limit, that is for v/c_{v} and v/c_{m} ≪ 1, where c_{v} is the velocity of light in free space and c_{m} denotes the phase velocity of a wave in the medium considered at rest, the electromagnetic field intensities can be expressed in terms of a pair of scalar and vector potentials by specifying a modified Lorentz condition. The time‐dependent Green's function associated with the hyperbolic partial differential equations satisfied by these potentials is determined explicitly.

Linear Representation of Spinors by Tensors
View Description Hide DescriptionA linear representation of spinors in n‐dimensional space by tensors is proposed. In particular, in three‐dimensional space a set composed by a scalar and a vector is associated to any two‐component spinor, while in four‐dimensional space the set corresponding to a four‐component spinor is composed by a scalar, a pseudoscalar, a vector, a pseudovector, and an antisymmetrical tensor of second order. The resulting formalism is then applied to Schrödinger's and Dirac's equations. In three‐dimensional space it turns out that the proposed procedure automatically assigns an intrinsic magnetic moment to an electron in a magnetic field without introducing any relativistic ideas or ad hoc assumptions. In four‐dimensional space we can write the Dirac equation in a generally covariant fashion, without introducing new concepts with respect to the usual tensor analysis. The zero‐mass Dirac equation splits into two sets of equations, describing respectively the neutrino and the photon. The possible bearing of the proposed approach upon the theories of elementary particles is briefly discussed.

Stability of Matter. I
View Description Hide DescriptionThe stability problem of a system of charged point particles is discussed, and a number of relevant theorems are proven. The total energy of a system of N particles has a negative lower bound proportional to when no assumption is made on the statistics of the particles. When all particles belong to a fixed number of fermion species, a lower bound exists proportional to N.

Translational Invariance Properties of a One‐Dimensional Fluid with Forces of Finite Extent
View Description Hide DescriptionThe translational invariance properties of a one‐dimensional fluid with finite range forces are investigated. For N particles in the interval [0, L], with a two‐body interaction potential w(x) = 0 for x ≥ R, we find the following: (a) If w(x) has a hard core of diameter d and R ≤ 2d, each n‐particle distribution functionD_{n} (x _{1}, …, x_{n} ) is translationally invariant if and only if L > 2(N − n)R and x _{1}, …, x_{n} lie in [(N − n)R, L − (N − n)R]. (b) For arbitrary finite values of R, with or without a hard core, the above conditions are sufficient for translational invariance of the D_{n} . These conditions hold for all temperatures.

Symmetry Group of the Hydrogen Atom
View Description Hide DescriptionIt is shown how the complete dynamics of the hydrogen atom is related to the three‐dimensional rotation group.

Variational Theorem for Reduced Density Matrices
View Description Hide DescriptionThis paper discusses a theorem concerning the variational description of the eigenfunctions and eigenvalues of the two complementary reduced density matrices for a many‐particle system in a bound state.

Relativistic Partial Wave Analysis of Electron Scattering
View Description Hide DescriptionThe partial wave series for the scattering amplitude for high‐energy electron scattering is not uniformly convergent. The singularity responsible for the nonuniform convergence containing terms in (sin ½θ)^{2iα − 2}, (sin ½θ)^{2iα − 1}, and (sin ½θ)^{2iα} is separated from the rest of the series so that an accurate partial wave analysis may be carried out for any scattering angle.

Mixed Irreducible Representations for U _{3}
View Description Hide DescriptionIn this paper we study some properties of irreducible representations of the unitary group in three dimensions U _{3} with positive and negative indices. These representations are useful for the group theoretical classification of particle‐hole states in nuclear shell theory, as well as in elementary particle physics. We show how the rules for reducing the direct product of two given representations should be modified when we include positive and negative indices and we use these rules to obtain an algebraic expression for the irreducible representations contained in the direct product.

Density Fluctuations of a Fluid
View Description Hide DescriptionA proof is given, in the canonical and the grand canonical formalism, to show that the density fluctuations in a macroscopic region of a fluid are large in the two‐phase states and small in the one‐phase states. The definition of large and small density fluctuations is one used previously by Dobrushin in connection with the Ising model. The density fluctuations at the critical point are, using this definition, small if the critical isotherm has no flat portion.

Strong Coupling Limit in Potential Theory. I
View Description Hide DescriptionAnalytic properties of the Jost function in g, the coupling constant, are studied for potentials which are L ^{½}, i.e., , and have a negative power or exponential tail for large distances. For the bound state and scattering problems, it is found that the Jost function has exponential order ½ for large g, which implies that the scattering phase shift and the number of bound states in an attractive potential grow like g^{½} for large g. The latter result is the best possible and is a considerable improvement over earlier estimates.

Variational Principles and Weighted Averages
View Description Hide DescriptionAn examination of two functionals, which are in common use for making variational estimates of weighted averages, reveals that one may be preferred over the other in certain cases. In particular, for a positive‐definite self‐adjoint operator, the normalization‐independent functional always yields the better approximation to the stationary value.

Calculation of Intersect Distribution Functions in Small Angle X‐Ray Scattering
View Description Hide DescriptionAs has been recently suggested, many subjects in small angle x‐ray scattering theory can be discussed by using a function called the intersect distribution functionG(M), which gives an average value of the distribution of lines with length M which pass through a point in a particle and which also have both ends lying on the boundary of the particle. Some properties of the intersect distribution function for a plane lamina with a convex boundary are investigated. The calculation is found to require the use of a weighting factor which is expressible in terms of the function generating the boundary of the lamina. The relation between G(M) and the two‐dimensional characteristic function is given. The exact intersect distribution function is found for a circle, and an approximate calculation of G(M) is carried out for small M for an arbitrary plane lamina with a convex boundary.

Correlations in Ising Ferromagnets. I
View Description Hide DescriptionThe following results are proved for a system of Ising spins σ_{ i } = ±1 in zero magnetic field coupled by a purely ferromagnetic interaction of the form −Σ_{ i<j } J_{ij} σ_{ i }σ_{ j } with J_{ij} ≥ 0, for arbitrary crystal lattice and range of interaction: (1) The binary correlation functions 〈σ_{ k }σ_{ l }〉 are always nonnegative (〈 〉 denotes a thermal average). (2) For arbitrary i, j, k, and l, 〈σ_{ i }σ_{ j }σ_{ k } σ_{ l }〉 ≥ 〈σ_{ i }σ_{ j }〉 〈σ_{ k }σ_{ l }〉. Consequences of these results, in particular the second, are: (i) 〈σ_{ k }σ_{ l }〉 never decreases if any J_{ij} is increased. (ii) If an Ising model with ferromagnetic interactions exhibits a long‐range order, this long‐range order increases if additional ferromagnetic interactions are added. This last fact may be used to prove the existence of long‐range order in a large class of two‐ and three‐dimensional Ising lattices with purely ferromagnetic interactions of bounded or unbounded range.

Correlations in Ising Ferromagnets. II. External Magnetic Fields
View Description Hide DescriptionResults of a previous paper showing that 〈σ_{ k }σ_{ l }〉 and [〈σ_{ k }σ_{ l }σ_{ m } σ_{ n }〉 − 〈σ_{ k }σ_{ l }〉 〈σ_{ m }σ_{ n }〉] are always positive for a system of Ising spins σ_{ i } = ±1 coupled by a purely ferromagnetic interaction (〈 〉 denotes a thermal average) are extended to the cases where (i) certain spins are constrained to have the value +1 or (ii) the system is placed in an external (``parallel'') magnetic fieldH. The theorems thus obtained provide a simple proof of the existence of ``bulk'' values for 〈σ_{ k }σ_{ l }〉 and for 〈σ_{ k }〉; the latter is identical with the usual bulk magnetization per spin. The correlation functions 〈σ_{ k }σ_{ l }〉 are monotone nondecreasing in H for fixed temperature T. Both 〈σ_{ k }σ_{ l }〉 and 〈σ_{ k }〉 (and thus the bulk magnetization) are monotone nonincreasing in T for fixed H ≥ 0.