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Volume 17, Issue 11, November 1976

Expansion formulas and addition theorems for Gegenbauer functions
View Description Hide DescriptionWe give a systematic summary of the properties of the Gegenbauer functionsC ^{α} _{λ}(x) and D ^{α} _{λ}(x) for general complex degree and order, with emphasis on the functions of the second kind, D ^{α} _{λ}(x), and on results useful in scattering theory. The results presented include Sommerfeld–Watson type expansion formulas and two reciprocal addition formulas for the functions of the second kind.

Bifurcation of solutions with crystalline symmetry
View Description Hide DescriptionWe consider the BBGKY equation for the single particle probability density in a hard sphere system. We investigate whether there is bifurcation from the fluid phase to functions which have crystalline symmetry. We find that as the density of the fluid increases from zero, there is bifurcation in one, two, and three dimensions. The bifurcation is shown to be characteristic of metastability and in general it does not occur at the equilibrium coexistence of two phases. The direction of branching and the stability of solutions near bifurcation is also discussed.

A soluble dispersion relation for a three‐dimensional band structure
View Description Hide DescriptionWe show that an exact dispersion relation can be obtained for a cubic lattice made of spherically‐symmetric attractive potentials. This result is obtained in a limit case where the potentials have zero range and infinite intensity.

On zeroes of the pion electromagnetic form factor
View Description Hide DescriptionWe develop a general procedure for the location of possible zeroes of the pionform factor, which relies on interpolationtheory for analytic functions. The zeroes are confined (in the unit disk) to regions bounded by (real) roots of algebraic equations and by algebraic curves. These regions depend both on the interpolation data and the class of functions, which is suitable for the physical problem.

On intelligent spin states
View Description Hide DescriptionIn this paper we give a more compact representation of the intelligent spin states defined by Aragone, Guerri, Salamó, and Tani. Using this new representation, we discuss the differences between minimum uncertainty states, coherent Bloch spin states and intelligent states. The evolution of these states under a particular time dependent Hamiltonian is studied, showing the relevance of the noncompact subgroup K of the Lorentz group. Finally we analyze the radiative properties connected with the intelligent states for a pointlike medium. The main results are: (I) they have a nonvanishing dipole moment (as the Bloch states) and (II) the proper intelligent states give a spontaneous emission intensity which is different from the one provided by the Bloch states.

Higher indices of group representations
View Description Hide DescriptionThe nth order index of an irreducible representation of a semisimple compact Lie group,n a nonnegative even integer, is defined as the sum of nth powers of the magnitudes of the weights of the representation. It is shown, in many situations, to have additivity properties similar to those of the dimension under reduction with respect to a subgroup and under reduction of a direct product. The second order index is shown to be Dynkin’s index, multiplied by the rank of the group. Explicit formulas are derived for the fourth order index. A few reduction problems are solved with the help of higher indices as an illustration of their utility.

Conservation of charge and the Einstein–Maxwell field equations
View Description Hide DescriptionIn a space of four dimensions I determine all possible second‐order vector–tensor field equations which are derivable from a variational principle, compatible with the notion of charge conservation and in agreement with Maxwell’sequations in a flat space. The general solution to this problem contains the Einstein–Maxwell field equations (with cosmological term) as a special case.

Continuum calculus and Feynman’s path integrals
View Description Hide DescriptionAn operational calculus is set up with the specific aim of resolving the problem of the integral of functionals in a general complex Banach algebra. The functionals that occur frequently in physics are Feynman’s path integrals for quantum mechanics which usually appear in the form of an exponential of an integral. Through the establishment of two operations, the r differentiation and pintegration, we succeed in constructing a formula, in closed form, for the integrals of Feynman type functionals. Applications to known problems (quantum harmonic oscillator, the electron–phonon system) corroborate conventional results. This formula is found to be at once consistent and more general than the method of projection into cylinder functionals of Frederichs type. It does not require an additive Gaussian measure, and it admits integration with finite limit functions. The methodology developed is general and applicable to other branches of mathematics. It is particularly suited to the study of infinitely divisible distributions in probability theory. We rederive, with facility, Lévy’s formulas for continuous sums, and Lévy–Khintchine and Kolmogorov formulas. We also find it applicable to continuum matrix algebra, where the formula for the determinant of matrices of continuous indices is given as a pintegral. As to algebraic identities, we give a continuum version of the binary expansion, and retrieve Stirling’s formula of factorials by pintegration. The i d e‐c l e f lies in the concept of infinitesimal ratio of a function in the same way that differential calculus deals with infinitesimal differences. Then the functional integral appears to be a natural product of the interaction between the conventional integration and the proposed pintegration. It also heralds the possibility of a generalized measure theory for integrals where the basic operation between the measure and the integrand is not bilinear.

On the irreducible representations of the Lie algebra chain G _{2}⊇A _{2}
View Description Hide DescriptionIn the first part of this article we solve the ’’state labelling problem’’ for the irreducible finite dimensional representations of the G _{2}⊆A _{2} chain, using a method applicable to other algebras‐subalgebras chains. In the second part we define, for these representations, operators analogous to those introduced by Nagel and Moshinsky for the A _{ n }⊆A _{ n−1} chain and explicitly construct the representations belonging to two equivalent classes.

Nonlinear response of equilibrium strongly coupled Fermi fluids. I. Formal development
View Description Hide DescriptionThis is Paper I of a series of three papers in which a self‐consistent propagator resummation of self‐energy effects in a strongly coupled Fermi fluid, in the presence of an external magnetic field, is performed. In the present paper, an exact expression is obtained for the grand potential in the presence of an external magnetic field which has a constant and a spatially varying part. The grand partition function and grand potential are written in terms of antisymmetrized cluster expansions. The cluster functions are then expanded in terms of a binary expansion. And, finally, an expression for the grand potential is obtained in terms of the reaction matrix for two‐body scattering and in a form suitable for the subsequent propagator resummation.

Nonlinear response of equilibrium strongly coupled Fermi fluids. II. Fourier expansion and partial resummation of expectation values—polarization diagrams
View Description Hide DescriptionThis is Paper II of a series of three papers in which a self‐consistent propagator resummation of self‐energy effects in a strongly coupled Fermi fluid is performed. In the present paper, a generating function for the expectation values of arbitrary one‐ and two‐body operators is introduced and written in the form of a cluster expansion. An explicit expression is written for the magnetization of a strongly coupled Fermi fluid in the presence of a constant and a spatially varying external field, and rules are given for evaluating it in terms of a reaction matrix expansion. A Fourier expansion is performed on a subclass of the diagrams contributing to the grand potential and the magnetization, and a self‐consistent resummation of self‐energy effects due to both the medium and the external spatially varying field is performed. It is found that traditional perturbation theory techniques for summing self‐energy effects cannot be applied to all terms in a reaction matrix expansion. The effect of the external fields on the polarization diagrams is discussed.

Propagator techniques for equilibrium strongly coupled Fermi fluids
View Description Hide DescriptionThis is Paper III of a series of three papers in which a self‐consistent propagator resummation of self‐energy effects in a strongly coupled Fermi fluid is performed. In the present paper, a Laplace transformation of the reaction matrix expansion of the grand potential and the magnetization is obtained, and a self‐consistent propagator resummation of self‐energy effects due both to the medium and to constant and spatially varying external magnetic fields is performed. The procedure for resumming polarization diagrams is discussed.

A calculation of SU(4) Clebsch–Gordan coefficients
View Description Hide DescriptionAll the Clebsch–Gordan coefficients for SU(4) that would be required for particle physics, including those decomposed with respect to SU(3), are obtained by using the general formalism of Baird and Biedenharn.

Green’s functions for a face centered orthorhombic lattice
View Description Hide DescriptionDiagonal and off‐diagonal matrix elements of the Green’s functions for a f a c e c e n t e r e d o r t h o r h o m b i c lattice are presented in terms of integrals of complete elliptic integrals of the first and third kind. These Green’s functions are also applicable to structures like that of the benzene crystal (space group D ^{15} _{2h }, interchange symmetry D _{2}).

Automorphisms of the Lie algebra of polynomials under Poisson bracket
View Description Hide DescriptionBy constructing the one‐parameter group of automorphisms generated by a typical derivation and generalizing certain special cases arising, we find all the automorphisms of the Lie algebra of polynomials under Poisson bracket. We introduce the notion of quasi‐Hamiltonian equations, and investigate the effect of transformations (q,p) → (α (q),α (p)) (α an arbitrary automorphism) on such equations. By considering linear quasi‐Hamiltonian equations with constant coefficients we obtain a conserved quantity for an a r b i t r a r y (2 × 2) linear system with constant coefficients.

On the linear connection and curvature in Newtonian mechanics
View Description Hide DescriptionThe trajectories of a scleronomic, holonomic particle motion in an otherwise general force field are autoparallel curves in a linear connected, symmetric, ’’almost’’ semimetric space. The Riemann–Christoffel curvature tensor and its concomitants belonging to the dynamical affinity are defined, and the physical meaning is discussed.

Tensor spherical harmonics and tensor multipoles. II. Minkowski space
View Description Hide DescriptionThe bases of tensor spherical harmonics and of tensor multipoles discussed in the preceding paper are generalized in the Hilbert space of Minkowski tensor fields. The transformation properties of the tensor multipoles under Lorentz transformation lead to the notion of irreducible tensor multipoles. We show that the usual 4‐vector multipoles are themselves irreducible, and we build the irreducible tensor multipoles of the second order. We also give their relations with the symmetric tensor multipoles defined by Zerilli for application to the gravitational radiation.

Solution of the multigroup transport equation in L ^{ p } spaces
View Description Hide DescriptionThe isotropic multigroup transport equation is solved in L ^{ p }, p≳1, for both half range and full range problems, using resolvent integration techniques. The connection between these techniques and a spectral decomposition of the transport operator is indicated.

On the tensor representation for compact groups
View Description Hide DescriptionA recent paper of Kasperkovitz and Dirl [J. Math. Phys. 15, 1203 (1974)] concerning the tensor representation for compact groups is examined critically. The flaw which is found in the main theorem fortunately does not affect the deductions which are made from that theorem.

Quantum field theory on incomplete manifolds
View Description Hide DescriptionA theory of the scalar quantum field on static manifolds is constructed using the language of Feynman Green’s functions. By means of examples in which the manifolds are parts of Minkowski space, we show how the ’’method of images’’ can be used to solve for the Green’s functions. In particular, we consider the Rindler wedge and the space outside a uniformly accelerated conducting sheet. As an example in which the manifold is nonstatic, we consider the region exterior to a conducting sheet which is accelerated impulsively from rest to the speed of light. Finally, we study the steady‐state part of de Sitter space where we do not obtain a unique result.