Volume 7, Issue 10, October 1966
Index of content:

Construction of Vanishing Cycles for Integrals over Hyperspheres
View Description Hide DescriptionThe homological methods of Fotiadi et al. are applied to the study of the analytic properties of an integral over a l‐sphere Γ of a closed meromorphic l‐form on the complex quadric of which Γ is the real section. Vanishing cycles are explicitly constructed at points of a certain standard type, and the relevant Kronecker indices are evaluated. The Picard‐Lefschetz theorem and the decomposition theorem are then applied to obtain linear relations between the discontinuities round various singularities. The results have a direct physical interpretation in the cases l = 4 and l = 2 in terms of the Riemann sheet structure of single‐loop Feynman diagrams with four, five, or six vertices. They give linear relations between the various discontinuity functions, which generalize the results obtained by Fotiadi and Pham for the two‐particle discontinuity of the five‐point loop and for the complete Feynman amplitude of the four‐point loop.

Variation of Green's Functions
View Description Hide DescriptionThe variation of the Green's function of a linear differential operator is computed as the variation of an n‐tuple integral with variable boundary. This generalization of Hadamard formula is shown to lead naturally to the method of ``invariant imbedding'' of R. Bellman. Three applications of the general formalism are given: the Dirichlet problem, the neutron or photontransport in a plane parallel anisotropicallyscattering slab, and scattering in a central field where three identities used in potential scattering are shown to be a consequence of the invariance of the asymptotic Green's function.

The Existence of Closed Magnetic Surfaces
View Description Hide DescriptionA toroidal vacuum magnetic field, which has closed magnetic surfaces in the neighborhood of the magnetic axis, is mathematically constructed. The existence of these surfaces is demonstrated by the theorem of Moser on the stability of mappings which are perturbations of rotations whose rotation angle is a function of the radius (twist mapping).

On the Dimer Solution of Planar Ising Models
View Description Hide DescriptionDerivations of the partition function of the Ising model on a general planar lattice L, which proceed via an associated dimer problem and use Pfaffians, are simplified by constructing a lattice L ^{Δ} (the ``terminal lattice'' derived from an ``expanded lattice'' of L) for which (A) the allowed dimer configurations are in one‐one correspondence with allowed Ising polygon configurations on L, and which (B) is planar if L is planar so that Kasteleyn's theorem may be used directly to construct the appropriate Pfaffian. This is in contrast to previous use of nonplanar associated dimer lattices for which the correspondence is not one‐one, so that is has been necessary to prove a somewhat obscure ``cancellation theorem.''

High‐Energy Behavior of Feynman Integrals with Spin. I
View Description Hide DescriptionA method of determining the leading behavior of planar graphs for two‐body processes in a spin‐½‐spin‐1 conserved vector currenttheory is outlined. The leading behavior can be found by inspection. Coefficients of lower‐order terms can be found explicitly. In a later paper we hope to use our methods together with analysis of nonplanar graphs to justify the Reggeization hypothesis in nth order.

On an Algebraic Method for the Decomposition of Direct Products of Representations of the Groups A _{2}(SU(3)) and B _{2}(SO(5))
View Description Hide DescriptionA prescription for the decomposition of the direct product of two irreducible representations of A _{2} and B _{2} is given, which is completely general and direct, i.e., it does not make use of any auxiliary means like Young tableau or Cartan‐Stiefel diagrams although it is based on the latter. The addition of weights alone under the observation of certain rules gives the desired result. This method can in principle be generalized to the semisimple groups of arbitrary rank provided an algebraic expression for the multiplicities of the weights contained in an irreducible representation can be found.

Scattering Cross Section for a Beam of Wave Packets
View Description Hide DescriptionThe scattering cross section is evaluated for a beam of wave packets of almost arbitrary form. It is found that this cross section is the superposition of the cross sections of various plane waves, which make up the wave packet without interference. For wave packets of well‐defined momentum, the usual rule that the scattering cross section is the absolute square of the scattering amplitude is regained.

Linked Cluster Theorem and the Green's Function Equations of Motion for a Many‐Fermion System
View Description Hide DescriptionThe equations of motion for the general many‐time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one‐ and two‐particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.

A Remark on a Reducible Quantum Field Theory with a One‐Parameter Symmetry Group
View Description Hide DescriptionThe connection between certain symmetry properties of a reducible algebra generated by the field operator and the decomposition into irreducible algebras is exhibited.

Classification of Irreducible Unitary Representations of Compact Simple Lie Groups. I
View Description Hide DescriptionAn irreducible unitary representation of any group belongs to one of the three Wigner classes, potentially real, pseudo‐real, or complex. The irreducible unitary representations of all the compact simple Lie groups except those of the type E are hereby classified. The similar classification for the simple groups E _{6}, E _{7}, and E _{8} is completed in the next paper.

Classification of Irreducible Unitary Representations of Compact Simple Lie Groups. II
View Description Hide DescriptionIn continuation to a previous article, the classification of the irreducible unitary representations of the groups E _{6}, E _{7}, and E _{8} into complex, potentially real, and pseudoreal catagories is completed.

Three‐Dimensional Formulation of Gravitational Null Fields. I
View Description Hide DescriptionThe four‐dimensional normal hyperbolic Riemannian space is represented as a direct product of a three‐dimensional space and a timelike line. The null gravitational field is defined in a manner analogous to that of electromagnetic field. It is shown that in this way three types of gravitational null fields can be characterized. We call them gravitational null fields of types A, B, and C. We find, as necessary and sufficient conditions, that the gravitational field be null field of types A and B, respectively. It is also shown that these null fields admit null vectors in accordance with the properties of gravitational radiation fields.

Bipolar Expansion of Screened Coulomb Potentials, Helmholtz' Solid Harmonics, and their Addition Theorems
View Description Hide DescriptionHelmholtz' solid harmonics (spherical Bessel functions × spherical surface harmonics) are generated from fundamental spherical waves through a ladder procedure using raising and lowering operators. The addition theorem for them and the bipolar expansion formula of a screened Coulomb potential are derived. A method for evaluating two‐center integrals with a general potential is given in the last section. This is new and useful in practical calculations. A key function,which appears in all the essential results, is studied in detail.

Discrete Degenerate Representations of Noncompact Rotation Groups. I
View Description Hide DescriptionThe discrete most degenerate principal series of irreducible Hermitian representations of the Lie algebra of an arbitrary noncompact as well as compact rotation group SO(p, q) are derived. The properties of these representations are discussed and the explicit form of the corresponding harmonic functions is given.

The Momentum Autocorrelation Function in a Bernoulli Chain
View Description Hide DescriptionThis paper is devoted to the study of the statistical dynamics of the small amplitude coplanar vibrations of a compound pendulum consisting of N + 1 particles suspended in series by weightless strings in a gravitational field. All particles have the same mass m, except for the top particle whose mass is m(1 + Q); and all strings are of equal length. The behavior of this system in the limit in which N → ∞ is of particular interest, because the maximum normal mode frequency is proportional to N ^{½}. In the limit N → ∞, asymptotic formulas with error estimates are obtained for the time dependence of the momentum autocorrelation function of: (1) the top particle when Q = 0; (2) the bottom particle when Q = 0; and (3) the top particle when N ≫ Q ≫ 1.

Some Remarks on the Construction of Invariants of Semisimple Local Lie Groups
View Description Hide DescriptionA general form of the l invariants of compact semisimple local Lie groups or rank l, as the traces of the powers of the ``velocity potential'' operator is suggested. The connection of this form of the invariants with those of Ref. 3 is described. The possible generalization beyond those of adjoint group and its connection with that of Biedenharn is discussed.