Volume 18, Issue 9, September 1977
Index of content:

Affine fields and the operator formulation of augmented scalar fields
View Description Hide DescriptionAffine fields, which can be used to replace the usual canonical fields, and which induce strictly homogeneous transformations of the underlying configuration space, are shown to be relevant in the operator formulation of augmented scalar field models. A characterization of the Hamiltonian and other basic generators by means of the expectation functional of the square of the field replaces the standard one based on the expectation functional of the field. Connection with previous work on augmented models is established through the form of the equation of motion for the field.

On generating forms of K‐generalized Lagrangian and Hamiltonian systems
View Description Hide DescriptionFrom a (n+1) ‐form Ω on the manifoldJ ^{ k } M of k‐jets of local sections of the vector bundle (M,π,N) we study the conditions to obtain the Lagrangian and Hamiltonian formalisms for a theory which involves higher order derivatives. The results generalize those of Gallissot and others for k=1.

On dual series equations involving Konhauser biorthogonal polynomials
View Description Hide DescriptionBy using Abel’s integral equations, we solve dual series equations involving Konhauser’s biorthogonal polynomial set of the first kind.

Equatorial circular geodesics in the Kerr–Newman geometry
View Description Hide DescriptionThe conditions for existence, boundedness, and stability are obtained for equatorial circular geodesics in the Kerr–Newman geometry. These conditions are extensions of the Bardeen e t a l. conditions in the Kerr geometry.

The Kirkwood–Salsburg equations for a bounded stable Kac potential. I. General theory and asymptotic solutions
View Description Hide DescriptionWe derive for Kac potentials of the form γ^{ s }ψ (λx) an expansion, for all the distribution functions, in powers of γ^{ s } (s=dimension) and prove for z<?_{cr} that the expansion is at least asymptotic. The coefficients in the expansion are shown to be solutions of linear operator equations similar to the Kirkwood–Salsburg equation. We also explicitly obtain a rather simple expression for the coefficients of γ^{ s } and show that they are given by solving the Ornstein–Zernicke integral equation with the choice of −βψ (y) for the direct correlation function.

The Kirkwood–Salsburg equations for a bounded stable Kac potential. II. Instability and phase transitions
View Description Hide DescriptionWe prove that systems interacting via potentials of the form φ (x_{1},x_{2}) =γ^{ s }ψ (γx _{12}) where ψ is bounded stable and defined on bounded support are unstable to fluctuations of wavenumber k′_{min}≠0 at a particular value v _{0} of v≡nβ, where n is the density and β=1/k _{ B } T in the limit γ→0 (VdW1). We also prove (in the VdW1) that the solution to the equation for the single particle distribution function bifurcates at this same value v _{0}, that the nonconstant solution is periodic and has a reciprocal lattice vector with a magnitude k′_{min}, and that there exists a type of long range order at v _{0}. These results are interpreted to indicate the existence of a spinodal point on the liquid isotherm, and similarities between this system and the known properties of the hard sphere fluid are discussed. A theorem is also proven about the range of activity where one has a unique fluid phase, and it is shown that this system has no coexistence region in the usual sense.

Lie theory and the wave equation in space–time. 5. R‐separable solutions of the wave equation ψ_{ t t }−Δ_{3}ψ=0
View Description Hide DescriptionA detailed classification is made of orthogonal coordinate systems for which the wave equation ψ_{ t t }−Δ_{3}ψ=0 admits an R‐separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have been found in previous articles. We find 106 new coordinates to give a total of 367 conformally inequivalent orthogonal coordinates for which the wave equation admits an R‐separation of variables.

Continuum calculus. II. The heterogeneous continuous functional differentiation applied to the Feynman path integral
View Description Hide DescriptionThe continuum calculus proposed previously [L. L. Lee, J. Math. Phys. 17, 1988 (1976)] is here extended to the study of continuous differentiation of a functional. The result, called HCFD, is shown to be the inverse operation of the functional integration for Feynman path integrals, in analogy to the case in ordinary differential calculus. The class of ’’separable’’ functionals is defined, which are useful in the derivation of the theory, playing a role similar to that of the characteristic functions in the Lebesgue theory of integration. A Radon–Nikodym type derivative is introduced in the definition of the continuous derivative for a general Banach algebra. This development constitutes a functional calculus of the continuum type. Comparisons with other types of functional derivatives are also made.

The topology of Euclidean Higgs fields
View Description Hide DescriptionIt is proved that in a class of Euclidean Higgs theories, a single index serves to describe both the gauge field and the Higgs field topologies.

Topology of Euclidean Yang–Mills fields: Instantons and monopoles
View Description Hide DescriptionWe discuss the topological properties of the Yang–Mills fields from a unified point of view of the Wu–Yang global formulation. In four‐dimensional Euclidean space, the gauge type is characterized by the second Chern class. The multi‐instanton solution recently found by ’t Hooft and generalized by Jackiw, Nohl, and Rebbi is discussed from this point of view. We also discuss the generalization of the topology of instantons and monopoles in higher dimensional spaces and the relation among them.

Notes on the symmetries of systems of differential equations
View Description Hide DescriptionThe concept of symmetry of the solutions of a system of differential equations is clarified. The functional character of the symmetry transformations is stressed in contrast with the pointlike character of the ordinary transformations considered by Lie. It is shown that any differential equation of arbitrary order possesses infinitely many symmetries, in strong contrast with a general theorem denying the existence of pointlike transformations of symmetry for an arbitrary differential equation of order greater than one. The relevance of local differential symmetries in theoretical mechanics is discussed, and some unsolved questions are raised.

Boost matrix elements and Clebsch–Gordan coefficients of the homogeneous Lorentz group
View Description Hide DescriptionIt is shown that the boost matrix elements of SO(3,1) obtained by Smorodinskii and Shepelev can be written as a sum of two Fourier series, whose coefficients are Clebsch–Gordan coefficients of SO(3) with complex angular momenta and magnetic quantum numbers. Moreover, the second term is equal to zero for all representations of the principal series except the most degenerate case. The connection between our expression and the spherical functions of Dolginov and other authors is explained. It is also noted that there are two ways of writing the boost matrix elements of SO(3,1), differing from each other by a phase factor. A proof for the orthogonality relation of the representation functions of SO(3,1) is given in the Appendix. The Clebsch–Gordan coefficients of SO(3,1) for the principal series in the general case (σ_{1}ν_{1}) × (σ_{2}ν_{2}) → (σ_{3}ν_{3}) are obtained as Xfunctions with complex angular momenta. An integral representation for these Xfunctions is obtained. It is shown that the CG coefficients of SO(3,1) have a multiplicity‐two problem. A solution to the multiplicity‐two problem is presented.

Conformal Killing tensors in reducible spaces
View Description Hide DescriptionIt is shown that the dimension of the vector space of second order, trace‐free conformal Killing tensors (CKT’s) in a Riemannian space of dimension n (?3) is bounded above by (1/12)(n−1)(n+2)(n+3)(n+4) and that this is attained in flat space. The discussion is eventually restricted to four‐dimensional spaces which admit a two‐dimensional, Abelian, orthogonally transitive symmetry group, as well as one nonredundant CKT. A sufficient condition is given for an empty space to be Type D.

Maxwell’s equations in an expanding universe
View Description Hide DescriptionSchrödinger’s classical solutions of Maxwell’sequations in an expanding universe with positive spatial curvature are reformulated in terms of gorup theory. Euler angles are used as coordinates in spherical space; the equations satisfied by the components of the complex electromagnetic field tensor are then given in terms of Euler angles. It is shown that if the fundamental modes of the electromagnetic field are appropriately chosen, certain components of the tensor are given by matrix elements of the irreducible representations of the group SU(2).

Fibre‐bundle structure of thermodynamic states
View Description Hide DescriptionIt is shown how the requirement that the Gibbs’ ensemble average 〈A〉=Tr(e ^{2H b } A)/TrA, b=−(1/2) k T, of any physical quantity A be formally expressible as an expectation value (P b‖A‖b P)/(P b‖b P) over a thermodynamic state ‖P b), naturally leads to the realization of (P b‖, as a cross section of a fibre bundle ϑ (G) with fibre G, over a manifoldM of pressure states P as b a s e space, where G is the infinite‐dimensional Lie group {exp(H b +u _{ n } P _{ n }); −∞<b, u _{ n }<∞} generated by H (Hamiltonian) and {P _{ n }; n=1,2,⋅⋅⋅}, the sequence of projectors on the eigenvectorsubspaces of H. The group G is thus partly parametrized by the temperature variable b.

On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations
View Description Hide DescriptionSolutions to the Cauchy problem for the equation are considered. Conditions on φ and F are given so that, for solutions with nonpositive energy, the following obtains: There exists a finite time T, estimable from above, such that as It is also shown that other ‐norms of a solution (including blow up in finite time.

Instantaneous Cauchy surfaces, topology change, and exploding black holes
View Description Hide DescriptionInstantaneous Cauchy surfaces are defined and several of their properties are given. Instantaneous Cauchy surfaces are achronal surfaces whose Cauchy development interiors are maximal on the set of all achronal surfaces. It is shown that topology changes in such surfaces always result in nonempty future Cauchy horizons and departures from global hyperbolicity. It is also shown that the structure which is usually assumed for the background spacetime of a self‐consistent exploding black hole implies a topology change in instantaneous Cauchy surfaces.

Pairing in the limit of a large number of particles
View Description Hide DescriptionExpressions for the ground and excited state energies of a system of nucleonsinteracting through pairing forces are given as a power series in inverse powers of the number of particles. The expressions are valid for systems with superfluidground states and either J=0 or L=0 pairing. The first three terms in the expansion are given explicitly, and they exhibit excitations with both vibrational and rotational (in isospin space) character. Analytical and numerical results are given for a model system with a two‐level single‐particle spectrum.

Twistors and induced representations of SU(2,2)
View Description Hide DescriptionWe give an explicit realization of a series of representations of SU(2,2) induced by R^{+}⊕SL(2,C). Vectors in these representation spaces are homogeneous spinor‐valued functions of two twistor variables. They may also be realized, in a frame‐dependent way, either as conformally invariant fields in Minkowski space or as homogeneous spinor‐valued functions on the O(2,4) null cone. The conformal invariance of the massless free fields is discussed from this point of view, and the twistor version of the field equations is derived. Finally, irreducible twistors are shown to correspond to conformally invariant fields satisfying the generalized twistor equation.

Post‐Newtonian two‐body and n‐body problems with electric charge in general relativity
View Description Hide DescriptionStarting with the Bażański two‐body post‐Newtonian Lagrangian with electric charge in general relativity, we construct a coordinate transformation (not involving center‐of‐mass coordinates) with two arbitrary parameters and obtain a Hamiltonian which is in agreement with one derived from quantum field theory. The field theory Hamiltonian corresponds to using an arbitrary parameter x _{ p } in the photon propagator as well as an arbitrary parameter x _{ g } in the graviton propagator. These results are also generalized to the case of n bodies. The condition for static balance e _{ i }=±G ^{1/2} m _{ i } is found to hold both for the exact Reissner–Nordstro/m ’’one‐body’’ problem and for the post‐Newtonian n‐body problem. An alternate condition for static balance e _{ i }=± (G m _{1} m _{2})^{1/2} is found to hold for the post‐Newtonian two‐body problem. The precession of the perihelion for the post‐Newtonian two‐body problem is given along with four special cases, one of which is the two‐body generalization of the ’’one‐body’’ special relativity result of Sommerfeld. Post‐Newtonian two‐body equations of motion (in center‐of‐mass coordinates) with the condition of static balance are also examined.