Volume 21, Issue 10, October 1980
Index of content:

Topological solution of ordinary and partial finite difference equations
View Description Hide DescriptionUsing the discrete path formalism, we obtain a topological solution for ordinary, as well as partial, linear inhomogeneous finite difference equations with variable coefficients and arbitrarily specified boundary conditions. The solution is homomorphic to a set of discrete paths constructed from a set of vectors determined by the level differences of the equation.

Finite nonabelian subgroups of SU(n) with analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients
View Description Hide DescriptionWe present two sequences of finite nonabelian groups which are semidirect products of three Z _{ n } groups. Although these groups are not simply reducible (in the tensor product of two irreducible representations an irreducible representation is obtained more than once) we give analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients.

Equivalence of induced representations
View Description Hide DescriptionEquivalence of induced representations for finite groups is considered in order to determine those equivalence classes of space group representations which are linked by complex conjugation.

Spinor fields invariant under space–time transformations
View Description Hide DescriptionSpinor fields i n v a r i a n t under the subgroups of the Poincaré group o r under the maximal subgroups of the conformal group of space–time are analyzed. It is shown that only certain Poincaré subgroups, all of dimension less than or equal to six, can leave two component spinor fields invariant, with rather severe restrictions on the fields. Tables listing all such invariant fields for subgroups of dimension greater than or equal to four are given. Construction of Dirac spinors and connections between invariant spinors and tensors are discussed: In particular it is shown that from any two‐component spinor invariant under a Poincaré subgroup a real skew‐symmetric tensor invariant under the same group may be constructed.

Oscillators submitted to squared Gaussian processes
View Description Hide DescriptionThe paper is a study of oscillators governed by equations of the type a Ẍ(t)+b Ẋ(t)+c X(t)=E(t), where a, b, c are given constants and where E(t) is for example the square of a Gaussian stationary process. A constructive and numerical method, using explicit expressions of Fourier transforms, are developed in order to compute the density of the distribution function (d.f.) of X(t) and of the joint distribution of X(t) and Ẋ(t). Hence the upcrossing rates of a given level and an approximation of the d.f. of max_{ t∈T } X(t) can be computed. A numerical example is given.

Factorization of operators I. Miura transformations
View Description Hide DescriptionThe method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is here extended to some third‐order scattering operators, and transformations between several fifth‐order nonlinear evolution equations are derived. Further applications are discussed.

A modified Bars–Durgut equation with polynomial eigenfunctions
View Description Hide DescriptionA singular integral equation in two degrees of freedom is defined. Its structure is similar to Bars–Durgut equation for baryons in two dimensions. It admits polynomial eigenfunctions and its spectrum can be studied exactly. A comparison with numerical data available for the baryonequation shows strong similarities. The asymptotic behavior of the eigenvalues for high quantum numbers is studied in the semiclassical approximation and it is found to be in good agreement with the exact spectrum. A peculiar feature of this model is the presence of a transition from a region of periodic classical orbits with constant frequency (straight Regge trajectories in the spectrum) to a regime of aperiodic orbits (nearly parabolic trajectories).

Variational principles for particles and fields in Heisenberg matrix mechanics
View Description Hide DescriptionFor many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one‐dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories,variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.

Dynamical symmetry and magnetic charge
View Description Hide DescriptionBy adding to the force between an electric and a magnetic point charge a central force arising from a specially chosen potential, one can construct a system known to have the same SO (3,1) and/or SO (4) dynamical symmetryalgebra as the Kepler system. We derive projective changes of variables under which the classical orbits of any such system are put in one‐to‐one correspondence with SO (3,1)‐ and/or SO (4)‐invariant sets of curves on similarly invariant surfaces. This extends results hitherto established only for the Kepler system. This is surprising in that there is a sense in which the phase space of such a magnetic system is a truncation of the Kepler phase space and so one might have expected such global properties not to generalize. Our transformations apparently do not permit transcription of the corresponding Schrödinger equation into a manifestly SO (3,1)‐ and/or SO (4)‐symmetric form, in contrast to the pure Kepler case. Such magnetic systems play roles in the theory of quantum fields in Taub–NUT space‐times, and in the theory of quantum‐mechanical fluctuations about extended magnetic monopoles in supersymmetric gauge theories. In passing, we use the properties of the magnetic systems to formulate a very short and direct proof that the classical orbits of the Kepler system are conic sections.

A factorization of M _{4}: Construction of the principal bundle of orthogonal frames over M _{4} from O(3,3) spinors
View Description Hide DescriptionThe principal bundle of orthogonal frames over M _{4} is explicitly constructed from certain pairs of Ō(̄3̄,̄3̄)́bar)= spinors that transform as (associated) twistors under the action of the covering group of the Poincaré group. In particular, flat space–time is constructed from these associated twistors, and is thus shown to be an object derivable from geometric structures more basic than the vectors of M _{4}. Associated twistors describe massive elementary particles. The position in M _{4} of such a particle can be explicitly defined in terms of the components of these twistors. The usual momentum and angular momentum variables which coordinatize the classical phase space of this elementary relativistic system of nonzero mass and arbitrary spin may also be realized in terms of this pair of associated twistors. This realization is not equivalent to descriptions of massive particles using twistors which have previously appeared in the literature.

Asymptotic behavior of gravitational fields in a type II coordinate system
View Description Hide DescriptionWith the aid of Penrose’s conformal technique the asymptotic behavior of the components of the metric tensor, the Weyl tensor, the Ricci tensor and the spin‐coefficients is calculated for a large class of space‐times that includes the NUT (Newman–Unti–Tamburino) solution as well as all asymptotically flat space‐times. The calculations are done in a coordinate system associated not with null hypersurfaces but with an asymptotically shearfree twisting null congruence. For vacuum the results presented here reduce to those of Aronson and Newman to the order given in their paper.

Space‐times with geodesic, shearfree, twistfree, nonexpanding rays
View Description Hide DescriptionThe Kundt class of metrics containing geodesic rays with vanishing divergence, shear and curl is obtained for more general Ricci tensors using the standard Newman–Penrose formalism. These solutions are then rederived using Penrose’s conformal technique, thereby clarifying their asymptotic behavior.

Lattice Green’s functions for cubic lattices
View Description Hide DescriptionThe Green’s functions for a cubic lattice given by G(E)=1/π^{3} ∫_{0} ^{π}∫_{0} ^{π}∫_{0} ^{π} (d x d y d z)/[E−ω (x,y,z)], where (i) ω(x,y,z)=(a _{1}cosx+a _{2}cosy)(1+cosz)+a _{3}cosz, (ii) ω(x,y,z)=a _{1}cosx(1+cosy+cosz+cosycosz) +a _{2}cosy+a _{3}cosz+a _{23}cosycosz are evaluated exactly and expressed as products of two _{2} F _{1}’s each of which represents a complete elliptic integral of the first kind. The expressions for the Green’s functions manifest the expected symmetries.

Covariant objects and invariant equations on fiber bundles
View Description Hide DescriptionLet P(M _{4},G) be a principal fiber bundle over the Minkowskian space–time M _{4} with the structural group G. The group G is supposed to be a compact and semisimple Lie group. Let A be a connection form on P(M _{4},G) and F=D A its curvature form. Let g _{ G } be the Cartan–Killing metric on G, and g _{ M 4 } the Minkowskian metric on M _{4}. Let us define dπ : T P→T M _{4}, the differential of the canonical projection from P onto M _{4}. Then we can define a scalar product for any two vectors from P(M _{4},G): g _{ P }(X,Y)=g _{ G }(A(X),A(Y))+g _{ M 4 } Q (dπ(X),dπ(Y)). In this metric the horizontal and vertical subspaces of the connection A are orthogonal to each other. Next, we construct the Clifford algebra corresponding to the metric g _{ P }. The metric g _{ P } can be always diagonalized locally to give diag((3+N)+,1−), where N is the dimension of G. The lowest faithful representation of this algebra, which we call C(3+N,1) is of the dimension K=2^{[(N+5)/2]}. This K‐dimensional vector space is called the space of spinors over P(M _{4},G). We study the decomposition of these spinors into multiplets of Lorentz spinors. We also define the generalized Dirac equation for such a spinor, construct an explicit representation in the case of G=SU(2), and give the formulae for the mass splitting. Finally, the invariant interaction with vector fields over P(M _{4},G) and scalar multiplets is discussed, together with the physical implications of the coupled equations.

S–matrix for interacting A–fields
View Description Hide DescriptionIn the framework of the Lagrangian field theory a new statistics for charged tensor fields is considered. An interaction Lagrangian is constructed such that the S–matrix is unitary, covariant and causal.

Unitarity of supergravity and Z _{2} or Z _{2}×Z _{2} or Z _{2}×Z _{2}×Z _{2} gradings of gauge and ghost fields
View Description Hide DescriptionUnitarity can be proven from the usual Z _{2} grading of gauge and ghost fields, or from a Z _{2}×Z _{2} grading, geometrically derived by Ne’eman and Thierry‐Mieg, or from a Z _{2}×Z _{2}×Z _{2} grading derived here. The claim that only the Z _{2}×Z _{2} grading leads to unitarity is incorrect. The opposite is shown to hold: signs due to different gradings are physically unobservable. We show how the Z _{2}×Z _{2} grading follows from the Z _{2} grading by taking a product space.

Space spinors
View Description Hide DescriptionJust as Maxwell’selectromagnetic field equations govern the evolution of electric and magnetic spatial vectors if some choice of time function has been made, so also the neutrinoequation and Dirac equation may be understood as governing the evolutions of certain spatial quantities. In this space‐plus‐time view of the spinor field equations, it is accurate and natural to regard a two‐component are written in 3‐plus‐1 form for both the spinor fields and the corresponding null vector fields. A spatial null vector is of the form M=E+iB, with E⋅E−B⋅B=0=E⋅B, so it is also of the correct algebraic form for describing a null electromagnetic field. The time derivative of a squared neutrino field M _{ a }, however, is ‐i curl M _{ a }+〈M〉^{ c } D _{ a } M _{ c }, compared with simply ‐i curl M _{ a } for a source‐free Maxwell field. Here 〈M〉^{ c } is the real spatial unit vector in the neutrino propagation direction E×B, and D _{ a } is the spatial covariant derivative.

The Dirac inverse spectral transform: Kinks and boomerons
View Description Hide DescriptionThe inverse spectral transform (IST) is derived when using the eigenvalue problem for the onedimensional Dirac operator: (D)=iσ_{3}(d/dx)+i(_{ r } ̃^{0} _{0} ^{q̃}), σ_{3}=(_{0} ^{1} _{1} ^{0}), where the potentials q̃ and r̃ have nonzero asymptotic values. The method used is of AKNS type. It is shown that the nonlinear evolution equations (NEE) obtained are of differential type at any order (and not of integrodifferential type). Some particular solutions are studied, and it is shown that their special behavior is a direct consequence of the nonzero boundary condition on (D).

Parametrizations of unitary matrices and related coset spaces
View Description Hide DescriptionExplicit forms of U3 and U4 matrices as functions of a single adjoint vector are displayed. Parametrizations of the coset spaces U(N+r)/(UN×Ur) are discussed, most explicitly for r=1 and 2, and related, for N=3 and 4, to the results for U3 and U4 matrices.

Field fluctuations in a two‐phase random medium
View Description Hide DescriptionWe consider here the problem of determining the mean square fluctuations in a statistically homogeneous isotropic two‐phase dielectric random medium. An expression is derived for a weighted sum of the mean square fluctuations in each phase in terms of the effective dielectric constant. From this expression bounds are derived for the mean square fluctuations in each phase. An assumption is then made to allow us to obtain exact expressions for the mean square fluctuations in a particular phase.