Volume 10, Issue 12, December 1969
Index of content:

Internal Multiplicity Structure for the Chain
View Description Hide DescriptionThe partition function for SU(n) is given in terms of that for SU(n − 1) through a recursion formula which is derived using the method of generating series. The usefulness of the expression is demonstrated in the cases of specific values of the rank.

Wavefunctions on Homogeneous Spaces
View Description Hide DescriptionThe properties of a class of homogeneous spaces of the Poincaré group are discussed. An 8‐dimensional space appears especially promising and the explicit unitary irreducible representations corresponding to physical particles are given using scalar wavefunctions on this space.

Treatment of Degeneracies in the Schrödinger Perturbation Theory by Partitioning Technique
View Description Hide DescriptionThe degenerate case in the Schrödinger perturbation theory has been treated by use of the partitioning technique developed by Löwdin. In order to simplify the concept and treatment, the repeated partitioning technique is utilized. This repeated partitioning allows us to use a one‐dimensional reference space and to determine the correct zero‐order wavefunction φ_{ A }.

Quasiparticle Formalism and Atomic Shell Theory
View Description Hide DescriptionThe quasiparticle formalism developed by Armstrong and Judd for atomic shells is extended to expose the complete group structure of the quasiparticle eigenfunctions of the equivalent electron l shell. A simple method for relating quasiparticle states to determinantal states and for calculating quasiparticle matrix elements is developed. The need for fractional parentage coefficients in calculating these matrix elements is entirely eliminated.

Character Analysis of U(N) and SU(N)
View Description Hide DescriptionA symmetric group analysis of the characters of U(N) and SU(N) representations yields formulas for (i) the multiplicities of weights in irreducible and tensor product representations, (ii) the coefficients occurring in the Clebsch‐Gordan series decomposition of Kronecker products with an arbitrary number of factors, (iii) the content of irreducible and tensor product representations of U(Σ_{ i } N_{i} ) with respect to representations of its direct product subgroup, , and (iv) the content of irreducible representations of U(NM) with respect to irreducible representations of . In particular, we exhibit formulas for (i), (ii), and (iii) containing only irreducible characters and Frobenius compound characters of the symmetric group. Under the application of an operator of the subgroup, with Σ_{ i } N_{i} < N, a vector in a representation of U(N) transforms as a linear combination of vectors in irreducible representations of the subgroup. We give formulas for determining the vectors occurring in such a linear combination. They are derived in a similar fashion to the formulas for (i), (ii), and (iii). In terms of weight diagrams, the formulas give the number of times a weight diagram of the subgroup's algebra occurs in the hyperplane generated by the application of the algebra to the weight of the U(N) vector in question.

Functional Integration and the Generalized Matthews‐Salam Equations
View Description Hide DescriptionVarious properties of Feynman functional integrals that appear in quantum field theory are studied. An indefinite functional integral is constructed. For the indefinite functional integral we prove a relation which is analogous in ordinary Riemann integrals to integration by parts. A special case of this relation gives an integration‐by‐parts formula for the Feynman functional integrals. In addition, various relations for integrating over variationals and variational derivatives are obtained. Application of these relations gives, among other things, a set of generalized Matthews‐Salam equations.

Application of Perturbation Theory to Many‐Body Systems with Localized Particles
View Description Hide DescriptionWe investigate the possibility of using perturbation theory to compute the binding energy for infinite systems in which the particles are localized. For the case of the linear chain of coupled harmonic oscillators, we prove that the perturbation series for the ground‐state energy per particle is convergent. Exact expressions for the generalized Padé approximants are derived. The generalized approximants provide a manifestly convergent sequence of approximations to the energy.

Physical Regions of Six‐Particle Processes
View Description Hide DescriptionThe physical regions of six‐particle processes are constructed in all planes of pairs of Lorentz‐invariant variables. As a matter of course, the permissible ranges of the eight independent variables are established. Thus, one application is the determination of the integration limits in phase‐space integrals that occur in calculations involving two‐to‐four and one‐to‐five particle processes.

Solution of a Three‐Body Problem in One Dimension
View Description Hide DescriptionThe problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is solved in one dimension.

Ground State of a One‐Dimensional N‐Body System
View Description Hide DescriptionThe problem of N quantum‐mechanical equal particles interacting pairwise by inverse‐cube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is considered in a onedimensional space. An explicit expression for the ground‐state energy and for the corresponding wavefunction is exhibited. A class of excited states is similarly displayed.

Self‐Consistent Approximations in Many‐Body Systems. II
View Description Hide DescriptionA stationary property of the grand‐canonical potential is introduced. This stationary property is used to define a class of self‐consistent approximations. The class of self‐consistent approximations is shown to be particularly suitable for the random‐phase approximation (RPA). The conditions for self‐consistency are only sufficient.

Vacancy Annihilation for One‐Dimensional Dumbbell Kinetics
View Description Hide DescriptionExpressions are developed which describe the ensemble average of the annihilation of groups of contiguous vacant compartments when spatially random attempts are made to place dumbbells on a linear array of N compartments. It is shown that in the limit, as the number of compartments tends to infinity, <θ_{ p }(t)>, the ensemble average of the fraction of the compartments which is composed of p contiguous vacant compartments, is given by,where v is the striking frequency of the dumbbells, t is time, and the C_{n} 's are appropriately defined coefficients.

Canonical States in Quantum‐Statistical Mechanics
View Description Hide DescriptionThe quantum‐mechanical analog of the classical Gibbs canonical density is characterized by considering a large collection Q of noninteracting quantum systems, each in an equilibrium statistical state. The set Q, the Hamiltonian operator for each system, and the statistical states are assumed to have certain properties which are given as axioms. It is shown that these assumptions imply that each member of Q is in a canonical state at a temperature which is the same for all systems. The possibility of zero absolute temperature is included.

Infinite Renormalization of the Hamiltonian Is Necessary
View Description Hide DescriptionWe show that the unrenormalized Hamiltonian in quantum field theory is unbounded from below whenever lowest‐order perturbation theory indicates that this is true. We conclude that perturbation theory is an accurate guide to the divergence of the vacuum energy in quantum field theory.

Relativistic Effects of Strong Binding on Slow Particles
View Description Hide DescriptionThe relativistic effect of strong binding on the equations of motion of slow particles is derived by taking the appropriate limit of classical relativistic equations of motion of interacting particles. The expected effect on the total mass of the system is verified. The relative motion is also affected‐in a modeldependent way.

Half‐Space Multigroup Transport Theory
View Description Hide DescriptionA method for solving various half‐space multigroup transport problems for the case of a symmetric transfer matrix is explained. This method is based on the full‐range completeness and orthogonality properties of the infinite‐medium eigenfunctions. First, the albedo problem is considered. A system of Fredholm integral equations is derived for the emergent distribution of the albedo problem, and it is shown that this system has a unique solution. Then, by using the full‐range eigenfunction completeness, the inside angular distribution is obtained from the emergent distribution. Finally, the Milne problem and the half‐space Green's function problem are solved in terms of the emergent distribution of the albedo problem and the infinite‐medium eigenfunctions.

Substitution Group and the Stretched Isoscalar Factors for the Group R _{5}
View Description Hide DescriptionThe phase relations for basis functions and Clebsch‐Gordan coefficients of the representations of the group R _{5} under the elements of the substitution group are given. The stretched isoscalar factors as well as the semistretched factors of the first kind are expressed in terms of the quantities of the theory of representations of SU _{2}.

Spectral Analysis of Classical Central Force Motion
View Description Hide DescriptionIt is shown here that the Liouville operator, which governs the development in time of a classical one‐particle system, has an absolutely continuous spectrum for a large class of attractive central force potentials. It follows that every absolutely continuous initial distribution of a monatomic ideal gas enclosed in a spherical container must approach a steady‐state distribution in time.

Feynman Path Integrals and Scattering Theory
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Kinematic Interaction for the Heisenberg Antiferromagnet at Low Temperature
View Description Hide DescriptionThe general theory of antiferromagneticspin waves is examined from a new viewpoint and the nature of the antiferromagneticground state is clarified. It is shown that the Anderson canonical transformation to free spin waves is nonunitary with the magnon state vectors having large nonphysical projections, and a projection technique which restores unitarity at low temperatures is developed. The nonphysical projection of the state vectors is shown to give a large kinematic interaction even at zero temperature.