Volume 9, Issue 9, September 1968
Index of content:

Exactly Solvable Many‐Boson Model
View Description Hide DescriptionA many‐boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N‐particle system is shown to be a product of N pair‐creation operators acting on the vacuum state. Each of these pair‐creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single‐particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two modelsystems. The first of these is one whose single‐particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single‐particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive‐interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single‐particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single‐particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V ^{⅔} in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single‐particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single‐particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.

Isotropic Solutions of the Einstein‐Liouville Equations
View Description Hide DescriptionThe gravitational fieldgenerated by a gas whose one‐particle distribution function obeys the Liouville equation is examined under the following assumptions: First, the distribution is locally isotropic in momentum space with respect to some world‐velocity field; second, if the particles have rest‐mass zero, the gas is irrotational. It is shown that the model is then either stationary or a Robertson‐Walker model. The time dependence of the radius in the Robertson‐Walker models is given in terms of integrals containing the distribution function.

Asymptotic Behavior of Stieltjes Transforms. II
View Description Hide DescriptionThe results of a previous paper concerning the asymptotic behavior of Stieltjes transforms for large z are extended to prove theorems which hold uniformly for all directions in the complex plane. Special additional assumptions, which hold for all sufficiently large values of the argument of the function whose transform is taken, are required to obtain these extended results.

One‐ and Two‐Center Expansions of the Breit‐Pauli Hamiltonian
View Description Hide DescriptionThe orbit‐orbit, spin‐spin, and spin‐orbit Hamiltonians of the Breit‐Pauli approximation are expressed in terms of irreducible tensors. One‐ and two‐center expansions are given in a form in which the coordinate variables of the interacting particles are separated. In the one‐center expansions of the orbit‐orbit and spin‐orbit Hamiltonians the use of the gradient formula reduces some of the infinite sums to finite ones. Two‐center expansions are discussed in detail for the case of nonoverlapping charge distributions. The angular parts of the matrix elements of these Hamiltonians are evaluated for product wavefunctions.

Some Variational Principles for Integral Equations
View Description Hide DescriptionComplementary variational principles are developed for the solution of Fredholm integral equations with symmetric positive‐definite kernels. In particular, the theory is applied to linear equations of the type,and bounds are obtained for . When λ is negative, the bounds are complementary upper and lower ones. When λ is positive, the bounds are one‐sided, but an improvement is made on a result of Strieder and Prager [J. Math. Phys. 8, 514 (1967)]. A condition given by these authors for the existence of bounds does not seem to be strictly necessary, and alternative conditions are derived. Systematic improvement of bounds by iterative and scaling procedures is discussed.

Antisymmetric Projection in the Approximation of No Spin‐Orbit Coupling
View Description Hide DescriptionThe explicit form of the projection operator for constructing antisymmetric wavefunctions for N fermions in the approximation of no spin‐orbit coupling is developed. Projection is applied within the one particle approximation. It is shown that if the orbitals associated with the minority spin can be completely expanded in terms of the orbitals associated with the majority spin, then the projected Hartree‐Fock scheme is completely equivalent to unprojected Hartree‐Fock theory. In the unrestricted case, deviations from this condition are not expected to be large, and integral properties such as energies calculated in the projected scheme should not be significantly different from unprojected results. However, for such properties as spin density at the nucleus in atoms or ions with nominally closed s shell, there may be significant differences between projected and unprojected schemes.

Irreducible Corepresentations of Groups Having a Compact Simple Lie Group as a Subgroup of Index 2
View Description Hide DescriptionFor all possible extensions of the compact simple Lie groupsG, such that G be a subgroup of of index 2, we determine those corepresentations of in which unitary operators correspond to the elements of the subgroup G, antiunitary operators to its coset. We find that any irreducible unitary representation of G can be extended to an irreducible corepresentation of in various ways summarized in Table IV.

Invariants of Nearly Periodic Hamiltonian Systems. II
View Description Hide DescriptionIn a previous paper the first few terms of the adiabatic invariant of a particular class of dynamical systems were found by solving Liouville's equation. The system considered was a periodic motion to which small perturbations were applied. The period of the unperturbed orbits was a constant and the perturbations were time‐independent. In this paper similar methods are used to find the invariant for the more general system, in which the period of the unperturbed orbits is a function of the coordinates and in which the perturbation varies slowly with time. The results are applied to a simple example, the Lorentz pendulum.

Asymptotic Estimates of Feynman Integrals
View Description Hide DescriptionIn this paper, we consider the problem of determining logarithmic, as well as polynomial, asymptotic estimates for certain convergent integrals containing parameters. We state and prove an asymptotic theorem which gives the logarithmic asymptotic behavior of a convergent integral where any subset of the parameters becomes large while the remaining parameters remain bounded. This theorem is then applied to the photon and electron self‐energy graphs of quantum electrodynamics.

Scalar Product for Harmonic Functions of the Group SU(2)
View Description Hide DescriptionA scalar product is defined which results in the single‐ and double‐valued spherical harmonics spanning a seminormed linear vector space that carries all of the irreducible unitary representations of the group SU(2). The possibility of defining such a scalar product was indicated in a previous paper. A Hilbert space is derived from the seminormed space through a further construction involving equivalence classes of vectors.

Analytic Renormalization
View Description Hide DescriptionRenormalized Feynman amplitudes are defined by a method of analytic continuation in subsidiary parameters. The results are shown to belong to the class of renormalized amplitudes defined by Boguliubov, Parasiuk, and Hepp.

Violation of the Quantum Ordering of Propositions in Hidden‐Variable Theories
View Description Hide DescriptionA general definition of hidden‐variable theories in terms of the dual structure of states and propositions is proposed. As a consequence of a theorem due to Zierler and Schlessinger, this definition implies a violation of the quantum ordering of propositions in the corresponding hidden‐variable theory. This violation is shown explicitly for the theory of measurement due to Bohm and Bub.

Lorentz Covariant Distributions
View Description Hide DescriptionTensor distributions of several four‐vector variables which transform according to a finite representation of the Lorentz group are considered. We give a canonical classification of all possible forms of such objects; this is used to show that, in the relevant cases, it is sufficient to regularize them with respect to the invariants that may be formed out of the variables to obtain analytic functions. We apply this result to Wightman functions, showing a result similar to a theorem proved in position space by Borchers under different assumptions.

``Haag Theorem'' for the Point‐Coupling Relativistic Lee Model
View Description Hide DescriptionWe prove that the free and total (renormalized) Hamiltonians exist as essentially self‐adjoint operators in the relativistic Lee model without cutoff. Nevertheless, their domains of definition only have the zero vector in common and thus the interaction Hamiltonian is meaningless (``Haag's theorem'').

Generalized Second‐Order Relativistic Wave Equations. I
View Description Hide DescriptionA covariant second‐order wave equation, free of subsidiary conditions, is deduced from the familiar linear relativistic wave equation for a free particle of arbitrary spin by use of the representation‐invariant Lie algebra of O(4, 1). The correspondence principle is used to interpret the physical content of this generalized equation, which explicitly admits zitterbewegung and implies an inverse spin dependence for the rest energy. Without further assumption this generalized second‐order equation is equivalent to the Klein‐Gordon equation for the particular Lie algebras of the Dirac and Duffin‐Kemmer rings. For higher spins the imposition of a subsidiary condition, understood via the Bargmann‐Wigner analysis, extends the equivalence with the Klein‐Gordon equation and explicitly displays the above mass spectrum.

Ensemble of Random Matrices with a Random Bias
View Description Hide DescriptionThe ensemble exp [−γ Tr (H − H _{0})^{2}], where the eigenvalues λ_{ n } of H _{0} are given by some distribution f(λ), is investigated. In particular, the limits of large and small γ for the orthogonal case are considered. Formal expressions are obtained for various distributions of the eigenvalues in the two limits. The approximation developed in the large‐γ limit is also applied to the thermodynamics of an incompletely specified system. Further, it is shown that this approximation is easily extended to include the unitary and symplectic ensembles.

Special Functions and the Complex Euclidean Group in 3‐Space. III
View Description Hide DescriptionThis paper is the third in a series analyzing identities for special functions which can be derived from a study of the local representations of the Euclidean group in 3‐space. Here identities are derived which relate Gegenbauer polynomials, Whittaker functions, Jacobi polynomials, and Bessel functions. Among the results are generalizations of the addition theorems for solid‐spherical harmonics and a group‐theoretic interpretation of the Maxwelltheory of poles.

Natural Boundary and Initial Conditions from a Modification of Hamilton's Principle
View Description Hide DescriptionIn a Hamiltonian variational formulation of a field theory, certain boundary conditions arise naturally whereas others arise as constraints on the admissible variations, and no initial conditions arise naturally. When the Hamiltonian formulation is used to obtain approximate solutions to boundary‐value problems, the approximating functions need not satisfy the natural conditions but must satisfy the constraint conditions. Although, in many instances, it is desirable for the approximating functions to satisfy the constraint—and even the natural—conditions, in other instances it is imperative that the approximating functions do not satisfy certain constraint conditions. A procedure is introduced for transforming Hamilton's principle so that the initial conditions and all conditions at boundaries and internal surfaces of discontinuity arise naturally and no constraint conditions are required. The transformation is effected by modifying the principle slightly, using Lagrange multipliers in the classical manner, and adding an appropriate initial‐value term to the Lagrangian. A particularly useful approximation technique is applied to a problem with an internal surface of discontinuity, and it is shown that the transformed principle can be used whereas the usual form of Hamilton's principle cannot. It is noted that the transformed principle has an important advantage over the method of least squares.

Quantum‐Ergodic Problem
View Description Hide DescriptionThe Gibbs approach to quantum dynamics is justified for nonrelativistic systems. It is shown that infinite‐time and microcanonical ensemble averages may be equated wherever the former exists and whenever a maximal set of constants of the motion can be determined, provided that the ensemble average is over states whose images under the maximal set are virtually stationary. The relevance of the demonstration for quantum‐statistical mechanics is discussed in the light of the ensemble formalism and the classical ergodic problem.

Iterative Solutions by Means of Trial Operators
View Description Hide DescriptionThe problem of finding a weighted average of an unknown solution to an inhomogeneous equation is examined. An analytic approximation technique is developed in terms of an iterative series involving a trial operator. By choosing the operator so that successive terms in the series vanish, one obtains a solution which has characteristics similar to variational solutions to the problem. The iterative approach has the added features of giving error estimates, the sign of the error, a testing ground for the quality of classes of trial operators or functions, and a possible means of determining upper and lower bounds to the exact result. Several examples are given for both self‐adjoint and non‐self‐adjoint systems. It is shown that the trial operator approach can give useful analytic approximations, with results which may be superior to variational calculations.