Volume 17, Issue 4, April 1976
Index of content:

The infinitely renormalized field in the scalar field model
View Description Hide DescriptionThe system consisting of a relativistic scalar boson field interacting with a single spinless nucleon with kinetic energy taken to be independent of momentum is studied in d space dimensions. The interaction Hamiltonian is taken to be H _{ I }(f) =F_{ R } ^{ d }φ_{ f }(x) ψ* (x) ψ (x) d x, where f is a momentum cutoff. The physical Hilbert spaceK corresponding to the case f≡1 in d space dimensions is discussed. The time smoothed nucleon annihilation operator is constructed as a closable operator on K. First order estimates are established for ψ (h) in terms of the local (in momentum space) number operators on K for the case d=3. It is shown that the union of the ranges of the adjoints ψ* (h) is dense in K. The one particle Hamiltonian is related to the nucleon creation operator on K.

SU(6) isoscalar factors for the product 405×56→56, 70
View Description Hide DescriptionSU(6) isoscalar factors for the product 405×56→56, 70 are calculated. SU(3) isoscalar factors for the products 27×10→10, 8 and 1̄0̄×8→8 are also tabulated.

Clebsch–Gordan coefficients of magnetic space groups
View Description Hide DescriptionWe have obtained sets of homogeneous linear equations in the Clebsch–Gordan coefficients for magnetic space groups in terms of the matrix elements of the irreducible representations of the little cogroup of the linear subgroup of index 2. Depending on the types of the co‐representations in the triple product, 18 cases arise. These 18 cases can be divided into six categories. We have given explicit forms for one case in each category and have indicated how the other cases are to be treated. The formalism has been developed for projective co‐representations so that both the vector and the spinor case can be treated.

Solutions of the three magnon bound state equation. I
View Description Hide DescriptionRecently several unphysical solutions of the three magnonbound stateequation for the isotropic Heisenberg Hamiltonian have been found, and one unphysical solution for the Hamiltonian with longitudinal anisotropy has been computed. Here we complete the work for such unphysical solutions for all anisotropy from the Ising to the isotropic Heisenberg limit by directly solving the integral equation. Two types of wavefunctions are constructed. The eigenvalue of the first type satisfies a cubic equation in general, but gives only a real root in the Ising limit and a pair of complex conjugate roots in the isotropic limit. The other type has a single eigenvalue; this one, previously known numerically, is shown to have a simple analytic form.

Solutions of the three magnon bound state equation. II
View Description Hide DescriptionA unified derivation of all the unphysical bound state solutions of the three magnonbound stateequation found so far is given by making a simple algebraic transformation of the variables in the equation.

Comment on the reduction of an important 9‐j symbol
View Description Hide DescriptionAn error in the formula for the reduction of an important 9‐j symbol is pointed out, and the correct relationship is given.

Application of coherent state representation to classical x ^{6} and coupled anharmonic oscillators
View Description Hide DescriptionThe problem of obtaining perturbative solutions to the nonlinear differential equations which describe the motion of the x ^{6} and quartically coupled oscillators is treated by the use of the well‐known coherent state representation. The results exhibit the basic qualitative features of nonlinearities and the characteristics of a coupled system in the weak coupling limit.

On the Majorana transformation
View Description Hide DescriptionSome properties of the Dirac equation in its four‐ and two‐component forms suggested by the Majorana representation of Dirac matrices are derived. Extension of the ideas to higher spin is also given.

Canonical transformations and phase space path integrals
View Description Hide DescriptionA previous discussion of canonical transformations and path integrals is extended to the phase space path integral method. Within this approach a broader class of canonical transformations can be introduced than within the Lagrangian approach, including coordinate transformations and essentially all infinitesimal tranformations.

Numerical experiments on the Calogero lattice
View Description Hide DescriptionThis paper presents the results of computer experiments performed on one‐dimensional, classical mechanical, N‐body systems whose point particles interact pairwise via the potential V (r) =a r ^{2}+b r ^{−2}, where r is interparticle distance and where a and b are positive constants. When each particle interacts with all other particles, the numerical experiments indicate that the system is mathematically integrable for either free‐end or fixed‐end boundary conditions. On the other hand, when each particle interacts with only its nearest neighbors, the computer detects a transition from near‐integrable to stochastic behavior again for either free‐end or fixed‐end boundary conditions. Our results thus support the conjecture that integrability is highly sensitive to changes in the total interaction potential but insensitive to modification of boundary conditions.

Stationary scattering for N‐body systems involving Coulomb potentials
View Description Hide DescriptionA stationary Hilbert spacescattering theory is derived for N‐body systems involving Coubomb‐like potentials. The derivation is based on stationary representations of the α‐channel renormalized wave operators, Ω^{(α)} _{±}, having the form Ω^{(α)} _{±}= s‐lim_{ε→+∞} W ^{(α)} _{±ε} P ^{(α)} F ^{(α)} _{±ε}, where W ^{(α)} _{±ε} P ^{(α)} are the stationary operators which form the basis of short‐range stationary scattering theory and F ^{(α)} _{±ε}* are appropriate stationary ’’renormalization’’ terms.

A class of field theories with unique well‐defined functional integration
View Description Hide DescriptionA class of field theories is considered where the codomain of the field is taken to be a compact Hausdorff topological groupG. The product space G ^{ R } ^{ n } is then a compact Hausdorff topological groupG ^{ R } ^{ n } and as such there exists a unique measure on this space.

Solution of the Dirac equation with Coulomb and magnetic moment interactions
View Description Hide DescriptionThe Dirac equation for a charged spin 1/2 particle with an anomalous magnetic moment in the Coulomb field is solved. A new phenomenon of formation of very narrow resonances of very high mass at small distances is demonstrated.

Comultiplicator of finite magnetic groups
View Description Hide DescriptionA matrix algebraic method for constructing the comultiplicator group for any finite magnetic group is given. Each element of this comultiplicator group corresponds to an inequivalent factor system of the magnetic group.

Spinor representations and projective factor systems of crystallographic point groups
View Description Hide DescriptionWe have given the correspondence between the factor system of the spinor representations of the different crystallographic point groups and the factor system given by Doering and Hurley. It turns out that for the groups C _{1}, C _{ i }, C _{2}, C _{ s }, C _{2h }, C _{4}, S _{4}, C _{4h }, C _{3}, C _{3i }, C _{3v }, D _{3}, D _{3d }, C _{6}, C _{6h }, C _{3h } the factor system of the spinor representations are associated with that of the vector representations.

Removal of the nodal singularity of the C‐metric
View Description Hide DescriptionThe charged C‐metric is transformed into another exact solution of the Einstein–Maxwell field equations corresponding to a massive charged particle accelerated by an electric field. When the appropriate equations of motion are satisfied, the nodal singularity associated with the C‐metric disappears.

Probability measures on fuzzy events in phase space
View Description Hide DescriptionThe notion of fuzzy sample point is introduced, and generalized probability measures on fuzzy events are defined. This leads to the concept of spectral measure on fuzzy events. It is shown that such measures can be associated with quantum‐mechanical states when the fuzzy elementary events are represented by Gaussian distributions on phase space.

Hilbert spaces of analytic functions and generalized coherent states
View Description Hide DescriptionGeneralized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.

Critical‐exponent inequalities in the Ornstein–Zernike theory
View Description Hide DescriptionA recent generalization of the classical Ornstein–Zernike theory of critical scattering is used to derive inequalities for the critical‐point exponents which characterize a first‐order phase transition. These inequalities for fluids are found to differ from the corresponding Buckingham–Gunton inequalities for Ising ferromagnets and the Josephson inequality.

L _{ p }‐space techniques in potential scattering
View Description Hide DescriptionWe consider potential scattering of structureless particles for potentials V (x) contained in certain L _{ p }‐spaces. In particular we study the compactness properties of K (t) =V ^{1/2} exp(−i H _{0} t) V ^{1/2}, H _{0} being the free Hamiltonian. By interpreting (z) =∫^{∞} _{0} d t exp(i z t) K (t), Imz⩾0, as a Bochner integral, we find the following property for the total cross section σ_{tot}(ω) (ω is the energy variable): If V∈L _{1}(R^{3}) ∩L _{ p }(R^{3}), 3/2⩾p⩾2, then ∫^{∞} _{ω0 } dω[ω^{1/2}σ_{tot}(ω)]^{ q } is finite for some suitable ω_{0}⩾0 and 2p/(2p−3) <q<∞.