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Previous Issue

Dec 1969

Volume 10, Issue 12, pp. 2129-2318


Internal Multiplicity Structure for the Chain SU(n)⊃SU(n−1)⊃…⊃SU(2)

D. Radhakrishnan

J. Math. Phys. 10, 2129 (1969); http://dx.doi.org/10.1063/1.1664812 (3 pages) | Cited 4 times

Online Publication Date: 4 November 2003

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The 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

Henri Bacry and Arne Kihlberg

J. Math. Phys. 10, 2132 (1969); http://dx.doi.org/10.1063/1.1664813 (10 pages) | Cited 24 times

Online Publication Date: 4 November 2003

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The 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

Jong H. Choi

J. Math. Phys. 10, 2142 (1969); http://dx.doi.org/10.1063/1.1664814 (7 pages) | Cited 20 times

Online Publication Date: 4 November 2003

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The 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 φA.

Quasiparticle Formalism and Atomic Shell Theory

M. J. Cunningham and B. G. Wybourne

J. Math. Phys. 10, 2149 (1969); http://dx.doi.org/10.1063/1.1664815 (7 pages) | Cited 3 times

Online Publication Date: 4 November 2003

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The 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)

Stephen Blaha

J. Math. Phys. 10, 2156 (1969); http://dx.doi.org/10.1063/1.1664816 (13 pages) | Cited 8 times

Online Publication Date: 4 November 2003

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A 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(Σi Ni) with respect to representations of its direct product subgroup, U(N1)⊗U(N2)⊗… ≡ ⊗i U(Ni), and (iv) the content of irreducible representations of U(NM) with respect to irreducible representations of U(N)⊗U(M). 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, ⊗i U(Ni) with Σi Ni < 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

Robert L. Zimmerman

J. Math. Phys. 10, 2169 (1969); http://dx.doi.org/10.1063/1.1664817 (10 pages) | Cited 1 time

Online Publication Date: 4 November 2003

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Various 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

C. C. Rousseau and D. P. Saylor

J. Math. Phys. 10, 2179 (1969); http://dx.doi.org/10.1063/1.1664818 (6 pages) | Cited 2 times

Online Publication Date: 4 November 2003

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We 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

Richard P. McNeil and Richard A. Morrow

J. Math. Phys. 10, 2185 (1969); http://dx.doi.org/10.1063/1.1664819 (6 pages) | Cited 1 time

Online Publication Date: 4 November 2003

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The 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

F. Calogero

J. Math. Phys. 10, 2191 (1969); http://dx.doi.org/10.1063/1.1664820 (6 pages) | Cited 473 times

Online Publication Date: 4 November 2003

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The 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

F. Calogero

J. Math. Phys. 10, 2197 (1969); http://dx.doi.org/10.1063/1.1664821 (4 pages) | Cited 231 times

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The 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

O. Shlidor and M. Revzen

J. Math. Phys. 10, 2201 (1969); http://dx.doi.org/10.1063/1.1664822 (4 pages)

Online Publication Date: 4 November 2003

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A 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

R. B. McQuistan

J. Math. Phys. 10, 2205 (1969); http://dx.doi.org/10.1063/1.1664823 (3 pages) | Cited 9 times

Online Publication Date: 4 November 2003

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Expressions 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, <θp(t)>, the ensemble average of the fraction of the compartments which is composed of p contiguous vacant compartments, is given by
math
,where v is the striking frequency of the dumbbells, t is time, and the Cn's are appropriately defined coefficients.

Canonical States in Quantum‐Statistical Mechanics

Robert E. Kvarda

J. Math. Phys. 10, 2208 (1969); http://dx.doi.org/10.1063/1.1664824 (6 pages)

Online Publication Date: 4 November 2003

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The 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

James Glimm and Arthur Jaffe

J. Math. Phys. 10, 2213 (1969); http://dx.doi.org/10.1063/1.1664825 (2 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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We 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

Amnon Katz

J. Math. Phys. 10, 2215 (1969); http://dx.doi.org/10.1063/1.1664826 (5 pages) | Cited 14 times

Online Publication Date: 4 November 2003

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The 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

S. Pahor and J. K. Shultis

J. Math. Phys. 10, 2220 (1969); http://dx.doi.org/10.1063/1.1664827 (7 pages) | Cited 4 times

Online Publication Date: 4 November 2003

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A 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 R5

S. J. Ališauskas and A. P. Jucys

J. Math. Phys. 10, 2227 (1969); http://dx.doi.org/10.1063/1.1664828 (7 pages) | Cited 13 times

Online Publication Date: 4 November 2003

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The phase relations for basis functions and Clebsch‐Gordan coefficients of the representations of the group R5 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 SU2.

Spectral Analysis of Classical Central Force Motion

Reese T. Prosser

J. Math. Phys. 10, 2233 (1969); http://dx.doi.org/10.1063/1.1664829 (7 pages) | Cited 1 time

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It 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

Donald Gelman and Larry Spruch

J. Math. Phys. 10, 2240 (1969); http://dx.doi.org/10.1063/1.1664830 (15 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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Abstract Unavailable

Kinematic Interaction for the Heisenberg Antiferromagnet at Low Temperature

D. C. Herbert

J. Math. Phys. 10, 2255 (1969); http://dx.doi.org/10.1063/1.1664831 (9 pages) | Cited 16 times

Online Publication Date: 4 November 2003

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The general theory of antiferromagnetic spin waves is examined from a new viewpoint and the nature of the antiferromagnetic ground 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.

Statistical Average of Product of Phase Sums Arising in the Study of Disordered Lattices. I

Priyamvada Sah

J. Math. Phys. 10, 2263 (1969); http://dx.doi.org/10.1063/1.1664832 (4 pages) | Cited 1 time

Online Publication Date: 4 November 2003

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A recurrence formula is established to evaluate the statistical average 〈S(k1)S(k2) … S(kn)〉 in the limit of N → ∞, where S(k) = ∑j = 1Neikxj,x1,x2,…,xN, denotes the position of atoms in a disordered lattice, under the condition that interatomic distances are statistically independent and have the same probability distribution s(r).

Continuous Representation Theory Using the Affine Group

Erik W. Aslaksen and John R. Klauder

J. Math. Phys. 10, 2267 (1969); http://dx.doi.org/10.1063/1.1664833 (9 pages) | Cited 36 times

Online Publication Date: 4 November 2003

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We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumann's work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self‐adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self‐adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self‐adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our kinematical group, we have developed continuous representations analogous to those Klauder and McKenna developed for the canonical group, and have shown that the former representations have almost all the desirable properties of the latter.

Dirac Formalism and Symmetry Problems in Quantum Mechanics. II. Symmetry Problems

J.‐P. Antoine

J. Math. Phys. 10, 2276 (1969); http://dx.doi.org/10.1063/1.1664834 (15 pages) | Cited 22 times

Online Publication Date: 4 November 2003

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The quantum‐mechanical formalism developed in a previous article and based on the use of a rigged Hilbert space Φ ⊂ H ⊂ Φ′ is here enlarged by taking into account the symmetry properties of the system. First, the compatibility of a particular symmetry with this structure is obtained by requiring Φ to be invariant under the corresponding representation U of the symmetry group in H. The symmetry is then realized by the restriction of U to Φ and its contragradient representation U in Φ′. This double manifestation of the symmetry is related to the so‐called active and passive points of view commonly used for interpreting symmetry operations. Next, a general procedure is given for constructing a suitable space Φ out of the labeled observables of the system and the representation U describing its symmetry properties. This general method is then applied to the case where U is a semidirect product G = T[squaredtimes]Δ, with T Abelian. Finally, the examples of the Euclidean, the Galilei, and the Poincaré groups are briefly studied.

Uniform Asymptotic Theory of Edge Diffraction

Robert M. Lewis and Johannes Boersma

J. Math. Phys. 10, 2291 (1969); http://dx.doi.org/10.1063/1.1664835 (15 pages) | Cited 34 times

Online Publication Date: 4 November 2003

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Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a high‐frequency asymptotic expansion. In problems involving screens with apertures, this asymptotic expansion fails at the edge of the screen and on shadow boundaries where the expansion has singularities. The uniform asymptotic theory presented here provides a new asymptotic solution of the diffraction problem which is uniformly valid near edges and shadow boundaries. Away from these regions the solution reduces to that of Keller's theory. However, singularities at any caustics other than the edge are not corrected.

Analytic Properties of a Class of Nonlocal Interactions. II

D. Gutkowski and A. Scalia

J. Math. Phys. 10, 2306 (1969); http://dx.doi.org/10.1063/1.1664836 (13 pages) | Cited 7 times

Online Publication Date: 4 November 2003

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A definition is given for the function Sl(k) in the complex l plane for a class of nonlocal interactions, called F, which has been considered in a previous paper [J. Math. Phys. 9, 588 (1968)]. The definition is obtained by means of a new class of nonlocal interactions, called G, for which the definition of Sl(k) in the l plane can be determined by a ``dynamical interpolation.'' The analytic properties of potentials of class G are studied. Then a suitable approximation is defined which allows us to approximate any potential of class F by means of potentials of class G. Comparing the analytic properties of potentials of class G which sufficiently approximate any given potential of class F, it is shown that their total scattering amplitudes can be made as near as we please to each other and that there exist Regge trajectories which can be made as near as we please to each other. With the given definition, Sl(k) turns out to be an analytic function in the complex l plane, but for a finite number of poles. Some general properties of the Regge trajectories are discussed and some examples are given.
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