Volume 4, Issue 8, August 1963
Index of content:

Expansions Associated with a Pair of Singular First‐Order Differential Equations
View Description Hide DescriptionThis paper reports on the continuation of a study of the spectral properties of a Dirac operator. The analytical methods developed by Weyl and Titchmarsh for the analysis of the Sturm‐Liouville equation are extended to the investigation of a system of two singular first‐order differential equations. Expansions associated with the system are established and a convergence theorem is presented.

Kinetic Equation for an Unstable Plasma
View Description Hide DescriptionA kinetic equation is derived for the description of the evolution in time of the distribution of velocities in a spatially homogeneous ionized gas which, at the initial time, is able to sustain exponentially growing oscillations. This equation is expressed in terms of a functional of the distribution function which obeys the same integral equation as in the stable case. Although the method of solution used in the stable case breaks down, the equation can still be solved in closed form under unstable conditions, and hence an explicit form of the kinetic equation is obtained. The latter contains the ``normal'' collision term and a new additional term describing the stabilization of the plasma. The latter acts through friction and diffusion and brings the plasma into a state of neutral stability. From there on the system evolves towards thermal equilibrium under the action of the normal collision term as well as of an additional Fokker‐Planck‐like term with time‐dependent coefficients, which however becomes less and less efficient as the plasma approaches equilibrium.

Regularity of the T Matrix in the Case of Dirac Potential Scattering
View Description Hide DescriptionApplying Hunziker's method to the case of Dirac potential scattering, we prove analyticity of the T matrix in energy and momentum transfer. Our conditions on the potential are somewhat weaker, and the domains of analyticity contain the known ones as special cases.

The Nature of the Axioms of Relativistic Quantum Field Theory. II
View Description Hide DescriptionThis paper continues the study of the nature and interdependence of the axioms of relativistic fieldtheory; attention is focused on the notion of relativistic invariance. The central result of the present paper is the derivation of necessary and sufficient conditions for a representation of the covariant free field to admit a unitary representation of the inhomogeneous Lorentz group associated with the field operator. It is shown that only the standard Fock‐Cook representation has this property. The relevance of the requirement that the Lorentz group is represented by a unitary family associated with the field operator is exhibited by an analysis of the covariant representations of Shale and Segal. These representations involve extremal states which are not pure, and group representations by intertwining operators.

Generalized ``Cross‐Correlation'' Field Quantities as Solutions of Field Equations
View Description Hide DescriptionIt has been shown by Wolf that, if a field amplitude obeys the wave equation, then a derived field quantity (the cross correlation or mutual coherence function), also obeys the wave equation. This result helps clarify the subject of partially coherent light; for example, a Huygens' principle for the propagation of intensity is a consequence. A generalization is presented: For that class of linear partial differential equations (p.d.e.'s) in which at least one independent variable, say t, does not appear in the coefficients, one can construct from a solutionf(P, t), (P representing collectively the other independent variables) a generalized ``cross‐correlation'' function F(P, t; P _{1}, t _{1}, P _{2}, t _{2} ⋯) satisfying the same p.d.e. as f(P, t); F is an integral over s of f(P, t + s) g (s, P _{1}, t _{1}, P _{2}, t _{2} ⋯), where (P _{1}, t _{1}, P _{2}, t _{2} ⋯) are values of (P, t); for particular choices of the highly arbitrary g, F can be, for example, (a) the usual cross‐correlation function, used by Wolf, (b) a higher‐order correlation function, (c) the Hilbert transform of f(P, t), and (d) derivatives of f(P, t). For linear p.d.e.'s in which two or more independent variables have the above property, higher‐dimensional or vector versions of F are obtainable. The existence of this (two‐fold) generalization of Wolf's result suggests the possibility of other physical applications.

Random Matrix Diagonalization—Some Numerical Computations
View Description Hide DescriptionNumerical results of Monte‐Carlo calculations of spacing and eigenvalue distributions for the invariant and independent Gaussian orthogonal ensemble of Hamiltonian matrices are presented. Many of the histograms should be useful for comparison with experimental data. A table of the first few moments of each distribution is given. For the spacing distributions, such moments are equivalent to spacing correlation coefficients, and hence these are also made available indirectly.

Real Unitary Representation of the Many‐Channel S Matrix for Complex l and E
View Description Hide DescriptionA representation is established for the multichannel S matrix in terms of a matrix functionA(l, E) and a scalar functionB(l, E), both holomorphic in the domain formed by the product of the whole finite l plane with the finite E plane, cut only along the left‐hand dynamical cut. The representation satisfies the known analytic properties of S(l, E), and also all the generalized unitarity conditions of Peierls, LeCouteur, and Newton. The reality condition on S for complex l and E is guaranteed by a simple condition on A and B.

Relativistic Potential Scattering
View Description Hide DescriptionThe scattering properties of the relativistic two‐body problem, governed by the Dirac equation, are investigated. It is shown rigorously that the associated Hamiltonians are self‐adjoint, that the associated wave operators exist, and that the scattering operator exists and is unitary, all under suitable conditions on the potential. These conditions on the potential are analogues of those required for the nonrelativistic two‐body problem governed by the Schrödinger equation.

Continuous‐Representation Theory. I. Postulates of Continuous‐Representation Theory
View Description Hide DescriptionIn a continuous representation of Hilbert space, each vector ψ is represented by a complex, continuous, bounded function ψ(φ) ≡ (φ, ψ) defined on a set of continuously many, nonindependent unit vectors φ having rather special properties: Each vector in possesses an arbitrarily close neighboring vector, and the identity operator is expressable as an integral over projections onto individual vectors in . In particular cases it is convenient to introduce labels for the vectors in whereupon each ψ is represented by a complex, continuous, bounded, label‐space function. Basic properties common to all continuous representations are presented, and some applications of the general formalism are indicated.

Continuous‐Representation Theory. II. Generalized Relation between Quantum and Classical Dynamics
View Description Hide DescriptionThis paper discusses an application to the study of dynamics of the typical overcomplete, non‐independent sets of unit vectors that characterize continuous‐representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one‐to‐one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum‐mechanical action functional with respect to a restricted set of unit vectors whose c‐number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single‐particle problems, but also to finite‐spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.

A Thermodynamical Limitation on Compressibility
View Description Hide DescriptionIn their theory of thermostatics, Coleman and Noll have obtained a convexity inequality which places restrictions on admissible stress‐strain functions for elasticmaterials. Here we show that for an arbitrary elasticmaterial in an arbitrary state of strain F, the general convexity inequality implies that the modulus of compression k obeys the inequality where p̄ is the mean pressure, i.e. minus one‐third the sum of the principal stresses. Here k is defined to be the derivative of p̄ with respect to the mass density along a deformation process representing a uniform expansion from the state F.

Study of Several Lattice Systems with Long‐Range Forces
View Description Hide DescriptionA number of one‐ and two‐dimensional Ising lattice systems with long‐range ferromagnetic interactions are studied. The theory introduces as basic variables stochastic fields acting at each site, but goes beyond Weiss mean‐field theory (or the Bragg‐Williams approximation) in giving a complete account of the statistics of these fields. A transition is manifest in these systems by a shift in the values of the stochastic fields which are important for the calculation of the partition function. Particular attention is devoted to the critical region where the range of significant stochastic fields broadens. The equation of state for the lattice gas corresponding to this model is of the van der Waals type. Comparison is frequently made between these results and the properties of an analogous one‐dimensional continuum system studied by Kac, Uhlenbeck, and Hemmer.

Irreducible Representations of Generalized Oscillator Operators
View Description Hide DescriptionAll of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para‐Bose or para‐Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three‐dimensional rotation group in the para‐Fermi case, and with the three generators of the three‐dimensional Lorentz group in the para‐Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.

Formulation of the Many‐Body Problem for Composite Particles
View Description Hide DescriptionThe many‐body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second‐quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.

Evaluation of Certain Sums Arising in Chemical Physics
View Description Hide DescriptionIn this paper we show that the Poisson transformation of finite sums of the formgenerally leads to a quickly convergent sum for problems of physical interest. Examples related to molecular orbitaltheory and lattice dynamics are discussed.