Volume 7, Issue 9, September 1966
Index of content:

Lattice Constant Systems and Graph Theory
View Description Hide DescriptionThe two principal systems of lattice constants that have arisen in the study of cooperative phenomena and related problems on crystal lattices are the strong (low‐temperature) and the weak (high‐temperature) systems. The two systems are defined in terms of the concepts of graph theory, and a general theorem relevant to cluster expansions is stated. The interrelation of the two systems is studied and exploited to derive configurational data for the face‐centered cubic lattice. All star graphs with up to seven points (vertices) or nine lines (edges) that are embeddable on the face‐centered cubic are described. A general classification of stars with cyclomatic index 3 is given.

Percolation Processes. I. Low‐Density Expansion for the Mean Number of Clusters in a Random Mixture
View Description Hide DescriptionA cluster expansion, valid at low densities, is derived for the mean number of clusters in a random mixture of sites or bonds on a graph. It is shown that only clusters without a cut‐point (stars) are required, and a number of general theorems for determining the weights are proved.

The Enumeration of Homeomorphically Irreducible Star Graphs
View Description Hide DescriptionIn expressing properties of interacting systems in the form of series, in many cases only summation over star graphs is involved. The identification and classification of such graphs is simplified by reducing them to homeomorphically irreducible stars. These graphs can be regarded as being all the possible different topological types of star. A method is described which has been used to produce all the different homeomorphically irreducible stars which have cyclomatic numbers ≤ 5. In particular, it has been established that there are 118 such graphs with cyclomatic number 5, 111 of which are planar. Diagrams of these graphs are appended, and a table of their k weights, which are needed for obtaining series for percolation processes, is also given. An extension of the method has been used to count the numbers of homeomorphically irreducible stars containing up to 8 vertices and 12 edges, a table of which is given.

On Ordering and Identifying Undirected Linear Graphs
View Description Hide DescriptionA general linear ordering relation is presented which completely orders any subset of the undirected linear graphs with the same number of vertices. The discussion is then specialized to the ordering and the identification of the unlabeled stars with no vertices of degree two, which faithfully represent the basic topologies of all stars.

Normal Thresholds in Subenergy Variables
View Description Hide DescriptionThe discontinuity across the normal threshold in a subenergy variable is evaluated by S‐matrix methods. The effect on the discontinuity of continuing past higher Landau singularities is discussed.

The Icecream‐Cone Singularity in S‐Matrix Theory
View Description Hide DescriptionThe icecream‐cone singularity is analysed by S‐matrix theory methods. It is found to appear only in that part of the physical region predicted by finite‐order perturbation theory. The corresponding discontinuity is derived, and the nature of the singularity is discussed.

Existence of Solutions of N/D Equations
View Description Hide DescriptionThe manifold of solutions of certain classes of N/Dequations is considered, under the restriction that the scattering amplitude have a uniform bound in all complex directions. Two cases are treated: (i) The Nintegral equation is Fredholm, but the associated homogeneous equation may have solutions; (ii) the equation has the marginally singular behavior characteristic of an asymptotically constant left‐hand cut discontinuity. The existence theorem in the latter case proceeds by the construction of the appropriate resolvent.

The r‐Particle Distribution Function in Classical Physics
View Description Hide DescriptionAn expression for the r‐particle distribution function (r ≥ 3) for classical gases in terms of the pair potential is converted to an expression in terms of the pair correlation function by graph‐theoretical means. The new formula involves a sum over all basic graphs with G‐bonds, having r root points, where G(r) + 1 is the pair correlation function.

N Degenerate Modes
View Description Hide DescriptionThe effect of a bilinear coupling in a system of degenerate modes is discussed. The normal mode frequencies are obtained for some simple couplings. If each mode is coupled to all the other degenerate modes, the maximum shift in normal mode frequency due to the coupling increases with the number of degenerate modes. Using the same methods, a many‐boson problem and a truncated version of it can be compared. The energyeigenvalues have the same relationship as the exact and first‐order perturbation excitation energies of the Bogoliubov model Hamiltonian for bosons.

Homogeneous Dust‐Filled Cosmological Solutions
View Description Hide DescriptionThe Einstein field equations with incoherent matter are discussed for the case of homogeneous space‐time, i.e., for metrics allowing a four‐parametric, simply transitive group of motions. It is proved that the only universes satisfying the above are those of Einstein, Gödel, and Ozsvath.

On Quantization in Space with Torsion
View Description Hide DescriptionSome elementary quantization questions are discussed under the hypothesis that physical space is a homogeneous space of constant torsion. In such a space, a freely propagating point particle appears to have extension in the sense that it obeys the Schrödinger equation of a symmetric top. Modifications of commutators and scattering amplitudes are given. The geometrical methods are based on the correspondence with the associated group‐space.

Analytic Properties of Resolvents and Completeness
View Description Hide DescriptionThe main purpose of this paper is to prove, by using a simple formal procedure, the completeness of the set of eigenstates of a non‐Hermitian operator whose resolvent satisfies certain physically plausible analytic properties. It is also shown how the obtained completeness relation can be related to the scattering solutions of the eigenvalue equation with extension to the multichannel case. All proofs are heuristic only.

Relativistic Hydromagnetic Waves and Group Velocity
View Description Hide DescriptionThe velocities of relativistic hydromagnetic waves in a compressible, perfect fluid of infinite conductivity are calculated in the framework of general relativity. In the absence of viscuous and Joule heat losses, the flow is isentropic, and, therefore, the wavesurfaces are propagated without change of shape. The velocities are first obtained in terms of the four‐vector magnetic field and then in terms of the three‐dimensional field. Several limiting cases are considered, and, in particular, it is shown under what conditions the expressions reduce to the nonrelativistic forms. Finally, the group velocities are calculated. The existence of a group velocity for such waves is based on the fact that the velocities exhibit a directional dependence. The group velocity in this case is of significance because it is the velocity of energy propagation, just as in the case when dispersion exists.

Local States
View Description Hide DescriptionA heuristic discussion is given of the preparation of states in finite space‐time regions. Some axioms concerning such states are derived from this discussion. It is shown that states requiring for their preparation a selection of events according to the outcome of a measurement will not be ``strictly local.'' Nonselective states will, however, be ``strictly local.'' The mathematical structure of such states is investigated.

Absorptive Parts and the Bethe‐Salpeter Equation for Forward Scattering
View Description Hide DescriptionWe have developed a Laplace transform approach to the Bethe‐Salpeter equation for absorptive parts of forward scattering amplitudes. The method appears direct and unsophisticated and is useful for computation. It is essentially identical to decomposition into four‐dimensional partial waves, but the inversion formula is more straightforward. We have obtained the high‐energy behavior of sums of several types of φ^{4} graphs in the weak‐ and strong‐coupling limits. These examples illustrate some general results we obtain for Mth order φ^{4} kernels. Specifically, the absorptive part behaves as s ^{ n } _{0}(log s)^{β} for high s; for weak coupling λ, n _{0} ∼ λ^{½M }, while for strong coupling in the ladder graph approximation, n _{0}/λ^{½} → 1/2π. We have also proven an interesting inequality related to absorptive parts. One of its corollaries is that uncrossed ladder graphs ``majorize'' crossed ones.

A Formalism for Generalized Quantum Mechanics
View Description Hide DescriptionA generalization of a non‐Hilbert‐space formalism, developed in two previous papers, is presented. In this generalization observables with a many‐dimensional spectrum are considered. Furthermore, the fundamental mathematical tools are generalized by introducing generalized functions instead of measures. It is then proved that a physical equivalence of the present formalism with the Hilbert‐space formalism can still be established.

Ordering Theorems
View Description Hide DescriptionVarious theorems related especially to time‐ordered products are proven. Applications to quantum field theory include particularly simple derivations of the off‐mass‐shell extensions of the S‐operator and of the interpolating field. The latter are a generalization of the results of Glaser, Lehmann, and Zimmermann.

Algebraic Aspects of Trilinear Commutation Relations
View Description Hide DescriptionAn attempt is made to derive from the formalism of Schwinger's action principle, in a more convincing manner than previously described, a set of trilinear equal‐time commutation relations which contains the commutation relations first discussed by H. S. Green as special cases. Matrix representations of field operators satisfying the trilinear commutation relations are considered. Two representations are explicitly discussed: a four‐dimensional and an eight‐dimensional representation. The representations considered and the bilinear equal‐time commutation relations between the associated ``component fields'' obeying ordinary statistics are specified by irreducible representations of an algebra which is suggested by the trilinear commutation relations. The component fields associated with the same representation and of the same spin differ from each other in their bilinear equal‐time commutation relations with other fields. This difference is reflected in the interactions into which the various fields can enter.

On the Existence of Field Theory. I. The Analytic Approach
View Description Hide DescriptionThe problem of existence of solutions to local field equations is studied. We set up the field equations so that the solutions correspond to fixed points of a mapping of the space of Green's functions into themselves. We attempt to use analytic methods to determine these fixed points, in particular, the contraction mapping principle. To do this we perform a rotation to Euclidean space from Lorentz space; in Euclidean space we prove the existence of solutions to a large class of approximating equations to the field equations, obtained by requiring the Green's functions to be zero if they have more than a certain number of external particles. By this method we prove that there is only the trivial zero solution to certain types of bootstrap equations. The contraction mapping theorem does not appear powerful enough to discuss the complete field equations.

The Structure of 4‐Spinors
View Description Hide DescriptionThe possible underlying spinor spaces used in the description of Dirac 4‐spinors are enumerated and classified within the framework of vector Clifford algebra. The relation with the usual matrix formulation is reviewed. Proper Lorentz transformation, space and time inversions are described in the Clifford algebra formalism. A representation‐independent definition of charge sign conjugation is given.