Volume 4, Issue 10, October 1963
Index of content:

Canonical Form of the Covariant Free‐Particle Equations
View Description Hide DescriptionThe canonical form of the covariant equations for free particles of nonzero rest mass is proposed to be taken as , instead of , as suggested by Foldy. The connection of our representation with the usual forms of the Dirac and the Klein‐Gordon (K‐G) equations are discussed, each feature being compared with the corresponding one in Foldy's case. The case of the Dirac equation is treated in some detail. A study of the infinitesimal operators of the Poincaré group and the transformation properties of the wavefunction and the polarization operator in our representation lead us to conclude that the choice of operators and the definition of spin states adopted by Iu. M. Shirokov in his study of the Poincaré group corresponds directly to our representation and the canonical form proposed, rather than that proposed by Foldy, as is sometimes supposed. It is also shown that the proposed canonical form corresponds to Wigner's unitary representation of the Poincaré group in terms of the little group of (κ, 0, 0, 0) (for κ > 0). In Appendix A, we give a brief outline of the decomposition of the direct‐product representation of the Poincaré group to bring out the special features that arise in our representation. In Appendix B, we compare in detail, for the case of the Dirac equation, our transformation with the well‐known Foldy‐Wouthuysen transformation. The case of zero rest mass has not been considered. Also the discussion of the position operators in our representation has been left aside and is to be taken up in a following article.

Relativistic Position Operator for Free Particles
View Description Hide DescriptionCertain definitions and derivations for the relativistic center of mass are considered and are related to the position operator corresponding to the representation proposed in a previous paper. It is shown that the classical definition of Pryce has indeed a more direct correspondence with our representation than the Foldy‐Wouthuysen representation, in spite of the fact that Pryce's particular method of symmetrization leads to the latter. In the process of derivation, the points of view of Synge, Pryce, and Shirokov are compared and the expressions for the velocity and the intrinsic angular momentum corresponding to the different definitions are given. In the Appendix, our operator is compared with that of Newton and Wigner for the case of spin ½.

Invariant Solutions of the Exact Bethe‐Salpeter Equation in the General‐Mass Case
View Description Hide DescriptionThe invariant solutions of the exact Bethe‐Salpeter equation are found in the case of nonzero exchanged‐meson mass. They are expressed as series of ``truncated vertex functions,'' which are certain combinations of the usual vertex functions. The eigenvalues of the coupling constant are determined by a transcendental equation, which is not solved explicitly.

Remarks on the Double Dispersion Approach to the Bethe‐Salpeter Equation
View Description Hide DescriptionThe following remarks are made on the applicability of the double dispersion approach to the Bethe‐Salpeter equation introduced previously. (1) Any invariant solution of the Bethe‐Salpeter equation in ladder approximation satisfies the double dispersion representation when the total energy‐momentum is spacelike. (2) There are some exceptional invariant solutions which are not given by the previous method in the equal‐mass case, but the existence of such solutions is very unlikely in the unequal‐mass case. (3) In the case of the general separated kernel the previous results give the correct solutions even if the kernel does not reproduce the double dispersion representation.

Local Commutativity and the Analytic Continuation of the Wightman Function
View Description Hide DescriptionIt is proved that the analytic continuation of the Wightman function, the vacuum expectation value of the product of field operators, due to local commutativity is single‐valued in the union of the extended tubes which correspond to the Wightman functions obtained by permuting the order of the field operators in the product, and that the extended tubes, the union of them and the intersection of any two are simply connected.

Complex Singularities in Production Amplitudes
View Description Hide DescriptionIt is shown for a simple model of production that Watson's scheme for final‐state interaction may be good even though the amplitude has complex singularities. The method used consists in continuing analytically partial‐wave dispersion relations. Stress is put upon the necessity of correctly choosing the cuts starting at the complex singularities. The safest choice in this approach is to take the cut along the line on which the series of partial waves diverges. A simple geometrical recipe is given to see the evolution of this line, the appearance of an anomalous threshold and a complex singularity.

Regge Poles in High‐Energy Potential Scattering
View Description Hide DescriptionHigh‐energy potential scattering based on the Schrödinger equation has been investigated using Regge poles. When the Watson‐Sommerfeld (W‐S) integral is directly applied in the left half‐plane, it is not useful at all in order to derive the scattering amplitude but gives a condition which governs the behavior of Regge poles. They are distributed with residues such that their contributions cancel against each other in the nonforward direction, provided that the integral at infinity is ignored. When Mandelstam's technique is used, the Legendre function of the first kind and the secant in the original W‐S integral are replaced by the Legendre function of the second kind and the cosecant, respectively. This enables one to sum over partial waves more easily than with the conventional series. In high‐energy potential scattering one cannot single out an individual Regge pole. The same conclusion follows when Khuri's representation is used.

Computation of Matrix Elements of Integrable Nuclear Two‐Particle Interaction Operators
View Description Hide DescriptionA general procedure is described for computing matrix elements of operators that occur in a realistic nuclear Schrödinger Hamiltonian, in a basis of orbitals with radial factors of the form N_{a}r ^{2na +la } exp (−γ _{a}r ^{2}), where n_{a} is an arbitrary nonnegative integer and γ_{ a } is an arbitrary positive number. The method is suitable for efficient large‐scale computation of these matrix elements, needed when orbitals of physical interest (such as Hartree‐Fock orbitals) are expressed as linear combinations of basis functions of the kind indicated. The analysis of matrix elements of a tensor operator provides a new method of reduction to linearly independent reduced matrix elements, the number of which is smaller than it is in the usual analysis. Thus, the number of independent parameters in the corresponding empirical theory of complex nuclear spectra is reduced. The specific operator forms considered here are those present in the asymptotic one‐pion exchange potential, except that the functional forms of the potentials V(r) are not prescribed beyond the requirement that they should be integrable.

Reduced Width Amplitude Distributions and Random Sign Rules in R‐Matrix Theory
View Description Hide DescriptionA multivariate reduced width amplitude distribution is derived from quite general assumptions of level independence and of functional form invariance of the distribution. The multivariate distribution reduces to the well‐known Gaussian singlet distribution (one degree of freedom) and moreover allows for the possibility of channel‐channel reduced width amplitude correlations. Such correlations, which cannot be represented by a singlet distribution, may be especially relevant to partial fission reduced width amplitudes. In particular, it is noted that in statistical theory complete correlation or anticorrelation between two random variables implies a linear relation between them. The multivariate distribution is then used as a basis for a more precise statement of the ``random sign'' rules of R‐matrix theory than hitherto given. It is pointed out that these rules imply a linear averaging over the multivariate reduced width amplitude distribution (in addition to the usual linear energy average) and are not merely consequences of the equiprobability of positive and negative signs for the reduced width amplitudes. The influence of the energy eigenvalue distribution and of variable reduced width amplitude statistics is discussed. It is emphasized that the multivariate reduced width amplitude distribution is relevant to any reactiontheory in which level widths are defined even though it is phrased here in the language of R‐matrix theory.

Reduced Width Amplitude Distributions for the Unitary Ensemble
View Description Hide DescriptionThe reduced width amplitude distributions for a system not invariant under time reversal (unitary ensemble) are derived. The form of the joint distribution function is compared with that of the orthogonal ensemble. The calculation of the reduced width channel‐channel correlation coefficient shows that only a positive width correlation is possible.

Invariant Operators of the Unitary Unimodular Group in n Dimensions
View Description Hide DescriptionAn elementary derivation is given of Biedenharn's construction of a complete set of independent invariants for the group SU(n). The basic tool is the mapping of the adjoint representation onto the linear space of generators in the defining representation. The trace of any algebraic function of the matrix thus associated is seen to constitute an invariant of the adjoint representation and yields by substitution an invariant operator. The independent invariants are recognized by their isomorphy to the invariant forms under the permutation group.

Energy‐Dependent Boltzmann Equation in Plane Geometry
View Description Hide DescriptionThe paper presents the general procedure of solving the energy‐dependent Boltzmann equation in plane geometry. The particular solutions are found and then it is proved that the general solutions can be formed by superposition of particular solutions. As an illustration, the fully degenerate kernel is considered in detail and a solution in a closed form is obtained.

Computer Studies of Energy Sharing and Ergodicity for Nonlinear Oscillator Systems
View Description Hide DescriptionWeakly coupled systems of Noscillators are investigated using Hamiltonians of the formwhere the A_{jkl} are constants and where α is chosen to be sufficiently small that the coupling energy never exceeds some small fraction of the total energy. Starting from selected initial conditions, a computer is used to provide exact solutions to the equations of motion for systems of 2, 3, 5, and 15 oscillators. Various perturbation schemes are used to predict, interpret, and extend these computer results. In particular, it is demonstrated that these systems can share energy only if the uncoupled frequencies ω_{ k } satisfy resonance conditions of the formfor certain integers n_{k} determined by the particular coupling. It is shown that these systems have Nnormal modes, where a normal mode is defined as motion for which each oscillator moves with essentially constant amplitude and at a given frequency or some harmonic of this frequency. These systems are shown to have, at least, one constant of the motion, analytic in q, p, and α, other than the total energy. Finally, it is demonstrated that the single‐oscillator energy distribution density for a 5‐oscillator linear and nonlinear system has the Boltzmann form predicted by statistical mechanics. Thus, these nonlinear systems are shown to have many features in common with linear systems. In particular, it is unlikely that they are ergodic. From the standpoint of statistical mechanics, it is argued that this lack of ergodicity may be a welcome feature.

Representation Theory for Nonunitary Groups
View Description Hide DescriptionThe representation theory for nonunitary groups is formulated following the same development used in the case of unitary groups, and the orthogonality relations for the corepresentation matrices are obtained. The general considerations of Wigner are used to show how one may obtain the irreducible corepresentations of a nonunitary group from the irreducible representations of its unitary subgroup.

Convergence of Fugacity Expansions for Fluids and Lattice Gases
View Description Hide DescriptionUpper and lower bounds are obtained for R(V), the radius of convergence of the Mayer expansion VΣ_{ l } b_{l} (V)z^{l} expressing the logarithm of the classical grand partition function for a finite volume V as a power series in the fugacity z. The particles in V interact only through two‐body forces whose potential φ(r) satisfies s ^{−1} Σ_{ i<j≤s } φ(x_{ i } − x_{ j }) ≥ const ≡ −Φ for all s, x_{1} ⋯ x_{ s }. The bounds arefor any l ≥ 2. For lattice gases the integral becomes a sum. The upper bounds, obtained from the theory of entire functions, include a subsequence converging to R(V) as l → ∞. The lower bound is obtained by using the Kirkwood‐Salsburg integral equation to calculate upper bounds on the b_{l} (V)'s and the coefficients in the fugacity expansions of the s‐particle distribution functions. For hard‐core potentials some of these bounds can be strengthened. For nonnegative potentials, 1/2b _{2}(V) is an extra upper bound on R(V). The radius of convergence of the infinite‐volume series Σ b_{l}z^{l} is shown to be at least lim_{ V→∞} R(V), with equality for nonnegative potentials.

Remarks on the Combinatorial Approach to the Ising Problem
View Description Hide DescriptionA new proof is given of a certain conjecture due to Feynman. This conjecture relates graphs and paths on a lattice and was first proved by Sherman. It is the key step in a particular method of obtaining Onsager's formula for the partition function of the two‐dimensional Ising model.

Approximate Evaluation of Feynman Functional Integrals
View Description Hide DescriptionBy considering a new set of functions for the integration domain, Feynman's functional‐integral representation of the propagation kernel is evaluated approximately for values of the time that are not too large. The approximation theory is applied to systems of uncoupled and strongly coupled oscillators.

New Small‐Angle X‐Ray Scattering Calculation
View Description Hide DescriptionA method due to Hardy for evaluating finite integrals is used to obtain the structure factors for several rotationally symmetric shapes. The factor for particles formed by joining paraboloids of revolution is shown to be proportional to a second‐order Lommel function of two variables.

Note on the Global Validity of the Baker‐Hausdorff and Magnus Theorems
View Description Hide DescriptionThe Baker‐Hausdorff theorem states that for two given elements x and y in an associative algebra, the equation e^{x}e^{y} = e^{z} has a solution z which lies in the Lie algebra generated by x and y. The Magnus continuous analog gives an exponential solution to a linear operator differential equation. Both theorems are valid globally for free Lie algebras of formal power series. For algebras that are not free, however, both theorems are locally but not globally valid. Some examples are given. Necessary and sufficient conditions for global validity are discussed. A superior representation in terms of a finite product of exponentials is also given.

Errata: Nonrelativistic S‐Matrix Poles for Complex Angular Momenta
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