Volume 8, Issue 6, June 1967
Index of content:

Branching Theorem for the Symplectic Groups
View Description Hide DescriptionA theorem analogous to the Weyl branching law for the unimodular groups is derived for Sp(2n).

Fresnel's Equation in General Relativity
View Description Hide DescriptionThe Fresnel equation is derived in general relativity using the classical method applied by Levi‐Civita in the study of a nonrelativistic theory of electromagnetic induction. For the description of the anisotropic medium the theory proposed by Quan is adopted. The study of the Cauchy problem is also presented and the convergence of results assures us that the equation proposed is the good one.

Derivation of the Principle of Compensation of Dangerous Diagrams
View Description Hide DescriptionThe principle of compensation of dangerous diagrams (PCDD) postulated by Bogoliubov to determine the coefficients in the canonical transformation to quasi‐particles in superconducting systems is derived from four different criteria (1) the expected number of quasi‐particles in the true ground state is a minimum, (2) the one‐particle density matrix and the two‐particle amplitude determined from the BCSground state are equated to the true ones, (3) the expectation value of an arbitrary operator is simplified by diagonalizing its quadratic part, and (4) the starting point for the dressing of the quasi‐particle is chosen in the most convenient way. The condition obtained for the PCDD is then expressed in terms of quasi‐particle Green's functions. The ladder diagrams are eliminated by examining an integral equation for the Green's function describing the creation of two quasi‐particles from the vacuum. Finally, the condition obtained here for the PCDD is compared with the condition obtained previously.

Deformation and Contraction of Lie Algebras
View Description Hide DescriptionThis paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it is shown that the only groups which can be contracted in the Poincaré group are SO(4, 1) and SO(3, 2).

Asymptotic Theory of Electromagnetic and Acoustic Diffraction by Smooth Convex Surfaces of Variable Curvature
View Description Hide DescriptionA general method is presented for obtaining successive terms in short wavelength asymptotic expansions of the diffracted field produced by plane acoustic and electromagnetic waves incident on an arbitrary smooth convex surface. By introducing the geodesic coordinate system on arbitrary surfaces of non‐constant curvature, both scalar and vector integral equations governing the surface fields are solved directly. The expressions for leading and second‐order terms in the asymptotic expansion of the diffracted fields are obtained explicitly and the differences between acoustic and electromagnetic creeping waves are shown.

Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs
View Description Hide DescriptionNormalized lowering and raising operators are constructed for the orthogonal group in the canonical group chain O(n) ⊃ O(n − 1) ⊃ … ⊃ O(2) with the aid of graphs which simplify their construction. By successive application of such lowering operators for O(n), O(n − 1), … on the highest weight states for each step of the chain, an explicit construction is given for the normalized basis vectors. To illustrate the usefulness of the construction, a derivation is given of the Gel'fand‐Zetlin matrix elements of the infinitesimal generators of O(n).

Decomposition of the Unitary Irreducible Representations of the Group SL(2C) Restricted to the Subgroup SU(1, 1)
View Description Hide DescriptionThe unitary irreducible representations of the group SL(2C) belonging to the principal series restricted to the subgroup SU(1, 1) are decomposed into a direct integral of unitary irreducible representations of SU(1, 1). The matrix elements of the unitary operator which performs the decomposition are given explicitly and used to obtain a relation between the matrix elements of the unitary irreducible representations of the groups SL(2C) and SU(1, 1). Similar identities between the matrix elements of nonunitary representations of these groups are obtained by means of analytic continuation. The relevance of these results to the theory of complex angular momentum and of high energy nearly forwardscattering is pointed out.

Mathematical Methods for Evaluating Second‐Order Three‐Body Interactions between Atoms or Ions with Gaussian Wavefunctions
View Description Hide DescriptionIt is shown that the integrals occurring in the expression for the interaction energy between three atoms or ions in second‐order perturbation theory with Gaussian wavefunctions can be reduced to single integrals of three different types. The first two types are erf x and erf ix functions, whereas the third type is a single integral of the error function which is easily evaluated by electronic computation.

Asymptotic Behavior of Stieltjes Transforms. I
View Description Hide DescriptionSeveral theorems are proved concerning the asymptotic behavior of Stieltjes transforms as z approaches infinity, in a sector of the complex z plane which does not include the cut in the transform. The asymptotic behavior of the transform is related to the asymptotic behavior, for large values of the argument, of the function whose transform is taken.

Probability Distribution of the Radius of Gyration of a Flexible Polymer
View Description Hide DescriptionVarious aspects of the mathematics of the probability distributionP_{N} (S_{x} ) of one component of the square of the radius of gyration of an ideal Brownian chain with N units are presented. A rigorous expression for P_{N} (S_{x} ) in the form of a contour integral is obtained. The resulting integral is written in terms of Tchebichef polynomials. Several rigorous and approximate results are obtained for both the limiting distribution (N infinite) and for finite N.

Criterion for Bose‐Einstein Condensation and Representation of Canonical Commutation Relations
View Description Hide DescriptionThe existence of a Bose‐Einstein condensation in an interacting many‐boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair‐Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose‐field operator(x), where(x) has nonvanishing ground state expectation value, in accordance with the usual c‐number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low‐density limit. Bose‐Einstein condensation exists when the over‐all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operator. Yet the c‐number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.

Renormalized Coupling Constants and the Inversion Problem
View Description Hide DescriptionIt is shown how, starting from the (experimental) knowledge of scattering phase shift, energies of bound states and renormalized coupling constants, one is able to determine completely the parameters of a quantum field theoretic model previously considered by the authors, the so‐called Dyson model. The very same conclusion holds for the case of potential scattering, which is also briefly considered.

Remarks on Relativistic Statistical Mechanics. I
View Description Hide DescriptionThe modifications introduced by the specific forms of relativistic dynamics of many‐particle systems are shown to give rise to a different (with respect to the nonrelativistic case) manner to set the problems involved in a tentative construction of relativistic statistical mechanics. Although the difficult problems of relativistic dynamics are not solved, it is possible to define relativistic generalizations of phase space, distribution functions, Gibbs ensembles, and average values. In particular, phase space is chosen for convenience and is no longer related (as is usually the case) to the ``initial data,'' whose nature is yet unknown. As a consequence, only those observables which depend on the variables characterizing phase space give rise to easily computed average values. However, it is possible to enlarge at will the basic phase space and to define subsequent densities from which average values may be calculated. [Example: The calculation of average values of observables needs only densities of the form and observables involving acceleration variables need the enlarging of phase space so as to include the latter. On this enlarged phase space, densities of the form may be defined and are used to compute average values, etc.] The notion of equilibrium is discussed and suggestions for reaching the solution of this unsolved problem are made.

Basic Algebra of Antilinear Operators and Some Applications. I
View Description Hide DescriptionThe primary aim of this work is to find the canonical (i.e., simplest) form of antilinear operators which is the analog of the diagonal form of linear ones, as well as to obtain that class of antilinear operators which corresponds to the class of normal, i.e., diagonalizable, linear ones. To achieve this aim two basic tools are used: polar factorization of an arbitrary antilinear operator into a linear, Hermitian, positive, semidefinite, and anti‐unitary operator Â_{a} = Ĥ _{1} Û_{a} = Û_{a} Ĥ _{2}, and representation of antilinear operators by antilinear matrices, which are products of a matrix factor transforming by unitary congruence transformations and the operation of conjugation which is the same for all antilinear operators and all bases. The canonical form is defined as the simplest form of the matrix factor. The criteria of simplicity are: a quasi‐diagonal form with smallest possible submatrices, a maximal number of zeros in them, and as many positive numbers as possible among the nonzero elements. It is found that the analog of the diagonal form of linear operators is the second‐order canonical form consisting of a diagonal part with nonnegative elements, which is as large as possible, and of two‐by‐two submatrices with zeros on the diagonal and with at least one positive element. The operators having this form are those whose polar factors can be simultaneously canonical, taking for the anti‐unitary factor, essentially the Wigner canonical form. These antilinear operators are called normal ones, and they can also be defined by the following relations between the polar factors: [Ĥ _{1}, Ĥ _{2}]_{−} = 0 and or by the single commutator, . A simple procedure to obtain the canonical form of a given normal antilinear operator is developed. A few applications of the results obtained are outlined. They belong to different fields such as electric network theory,quantum mechanics, and self‐consistent Hartree‐Bogoliubov theory.