Volume 23, Issue 1, January 1982
Index of content:

On the classification of Clifford algebras and their relation to spinors in n dimensions
View Description Hide DescriptionA classification of all the Clifford algebras is given in terms of Kronecker products of the quaternion and dihedral groups. The relationship to spinors in n dimensions is explicitly determined. We show that the real Clifford algebra in Minkowski spacetime is distinct from both the algebra of Dirac matrices and the algebra of Majorana matrices, and cannot be realized by the spinor framework. The matrix representations of Clifford algebras are discussed, and are utilized to give a classification of the real forms of Lie algebras. We are thus able to relate Clifford, Lie, and spinor algebras in an intrinsic geometrical setting.

Modified fourth‐order Casimir invariants and indices for simple Lie algebras
View Description Hide DescriptionThe fourth‐order indices for Lie algebras have been defined and studied by Patera, Sharp, and Winternitz. We show that it may be more convenient to modify the original definition and that the modified fourth‐order indices are intimately related to eigenvalues of symmetrized fourth‐order Casimir invariants. Explicit expressions for these quantities are given and we also find a quartic trace identity involving the generic element of these Lie algebras. We discuss the triality principle for the Lie algebraD _{4} in connection with identical vanishing of the modified fourth‐order index for this algebra.

Gribov ambiguities from the bifurcation theory viewpoint
View Description Hide DescriptionThe connection between singularities of the Faddeev–Popov determinant and the local gauge degeneracy is discussed. A criterion relating the two phenomena is given. It is proved that a whole neighborhood of the local gauge copy is filled with copies of transverse potentials.

Lippmann–Schwinger equations for a scalar macroscopic field in an anisotropic stratified medium
View Description Hide DescriptionWe develop a general approach to solve the transmission problem for a scalar macroscopic field in an anisotropic stratified medium. The method is based on a chainlike set of functional equations of the Lippmann−Schwinger type. A typical example of the fields under consideration is the wave field of a particle in the effective masstensor approximation.

Some topics in pre‐Hilbert space
View Description Hide DescriptionWe examine some questions concerning pre‐Hilbert spaces and operators defined in them. Some classical results, which hold true in Hilbert space, are extended, under particular conditions, to noncomplete space.

Symmetry of time‐dependent Schrödinger equations. II. Exact solutions for the equation {∂_{ x x }+2i∂_{ t }−2g _{2}(t)x ^{2}−2g _{1}(t)x −2g _{0}(t)}Ψ(x, t) = 0
View Description Hide DescriptionThe maximal kinematical algebra of the Schrödinger equation {∂_{ x x }+2i∂_{ t }−2g _{2}(t)x ^{2}−2g _{1}(t)x−2 g _{0} (t)}Ψ(x, t) = 0 is known to be the Schrödinger algebras_{1}. The kinematical symmetries are realized as first‐order differential operators in the space and time variables. A subalgebra G of s_{1} is chosen and from G and its invariants a complete set of commuting observables are constructed. The solution space of the Schrödinger equation is identified with the appropriate irreducible representation space of G. The wave functions, simultaneous eigenvectors of the compatible observables, are computed as explicit functions of space and time. The properties of a system with a potential V(x, t) = g _{2}(t)x ^{2}+g _{1}(t)x+g _{0}(t) are discussed.

Statistics with number operator C C*
View Description Hide DescriptionThe number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.

Ground state energy bounds for potentials ‖x‖^{ν}
View Description Hide DescriptionA theory is developed from which both upper and lower analytic bounds on Schrödinger eigenvalues can be obtained. We propose a recursion algorithm with which ground energies for certain potentials can be rigorously bounded to arbitrary precision. These analytic and numerical methods, together with existing techniques, are applied to the ground state problem for power potentials ‖x‖^{ν}, ν≳0.

The three magnon bound‐state equation in one dimension
View Description Hide DescriptionThe mathematical analogy of the three magnon bound‐state equation with other momentum‐space integral equations is studied. It is shown that a variable transformation, similar to Wick’s transformation in the Bethe–Salpeter equation, leads to alternative methods of complete analytic solutions of this equation.

On the structure of Coulomb‐type scattering amplitudes
View Description Hide DescriptionOn the basis of the Gell‐Mann–Goldberger formula for Mo/ller operators we introduce a natural splitting of the total scattering operator S into the pure Coulomb scattering operator S ^{c} plus a remainder S = S ^{c}−2πi T ^{sc}, implying a decompostion of the total scattering amplitude f into the Coulomb scattering amplitude f ^{c} and a remaining part f ^{sc} f(k,ω,ω′) = f ^{c}(k,ω,ω′)+f ^{sc}(k,ω,ω′). Concerning continuity properties, etc., of f ^{sc}, close similarities between f ^{sc} and ordinary short‐range amplitudes are proved. In particular, we introduce transition operators t ^{sc}(z) and show how to obtain f ^{sc} by an appropriate on‐shell limit, thereby avoiding the notion of so‐called Coulomb transition operators and the difficulties associated with them. Possible extensions of this approach to charged three‐particle systems are also sketched.

The equivalence of the Feshbach and J‐matrix methods
View Description Hide DescriptionIt is shown, by takcing P to be the projection operator on the subspace of function space in which the potential is truncated, that the exact solution of the scattering problem for the truncated potential using the Feshbach formalism is identical to the J‐matrix solution.

The asymptotic form of the continuum wavefunctions and redundant poles in the Heisenberg condition
View Description Hide DescriptionThe derivation of the Heisenberg condition is re‐examined to show why it is not an identity for potentials possessing redundant poles. Consideration of several such potentials for which exact solutions are known reveals that, in the process of taking an asymptotic limit, the usual derivation of the Heisenberg condition improperly neglects a set of terms. These terms are just those necessary to make the Heisenberg condition an identity; more importantly, it is demonstrated that these terms, providing information on the redundant poles, i.e., the sum of the residues of S(k) at the redundant poles, come from the asymptotic expansion of the continuum wavefunction. By this we are able to give the details of the nature of the asymptotic expansion of the continuum wavefunction and the information contained therein.

The quantum pendulum in the WKBJ approximation
View Description Hide DescriptionEigenvalues are determined for the plane pendulum problem by the WKBJ method in one‐ and four‐term approximations, and the results are compared. It is found that at high quantum numbers, the four‐term WKBJ approximation can yield eigenvalues of eight‐significant‐figure accuracy, but for low quantum numbers the results continue to be poor.

A new mathematical formulation of accelerated observers in general relativity. I
View Description Hide DescriptionInvariant methods of modern differential geometry are used to formulate exact closed form expressions for the coordinate velocity and coordinate acceleration of a geodesic particle in the tangent space of a general relativistic accelerating rotating observer. The observation of a general vector field is shown to be definable in two ways from presymmetry and covariance arguments. Our results for the parallel translation definition of observation are shown to subsume existing work in both special and general relativity on accelerated observers.

A new mathematical formulation of accelerated observers in general relativity. II
View Description Hide DescriptionThe observation of a general vector field based on exp_{*} is employed to obtain formulas for the coordinate velocity and coordinate acceleration of a geodesic particle. Our results are shown to reduce to those based on a parallel transport definition of observation in special relativity. In general relativity the difference between the expressions for the coordinate velocity and coordinate acceleration derived from the two definitions of observation is given in terms of the Riemann curvature tensor.

Vacuum handles carrying angular momentum; electrovac handles carrying net charge
View Description Hide DescriptionNonsimply‐connected spacetimes can have locally defined Killing vectors that are globally defined only up to sign (pseudovectors). We show the existence of asymptotically flat vacuum spacetimes which are axisymmetric (have a rotational Killing pseudovector), are topologically trivial outside a spatially compact region, and which nevertheless have nonzero angular momentum. An analogous construction establishes the existence of source‐free Einstein–Maxwell spacetimes which are topologically trivial outside a spatially compact region and which nevertheless carry nonzero net electric charge. The existence of such spacetimes leads to a new variant of the combined positive energy‐cosmic‐censorship conjecture: Given an asymptotically flat vacuum or electrovac initial data set which is axisymmetric and geodesically complete, the asymptotic mass, chargeQ, and angular momentumJ satisfy m?[Q ^{2}+(J/m)^{2}]^{1/2}.

The form of Killing vectors in expanding HH spaces
View Description Hide DescriptionThe Killing vector structure of those spaces of complexified general relativity known as expanding hyperheavens is investigated using the methods of spinor calculus. The Killing equations for all left‐algebraically degenerate Einstein vacuum spaces are completely integrated. Using the available gauge freedom, the resulting homothetic and isometric Killing vectors are classified in an invariant way according to Petrov–Penrose type. A total of four distinct kinds of isometric Killing vectors and three distinct kinds of homothetic Killing vectors are found. A master Killing vector equation is found which gives the form that the Lie derivative of the metric potential function W must take in order that it admit a given Killing vector.

An exceptional type D shearing twisting electrovac with λ
View Description Hide DescriptionA new electrovac with λ type D solution and with both Debever–Penrose vectors aligned along the real eigenvectors of the electromagnetic field is presented. The principal null directions are shearing and twisting. The existence of this solution, endowed with an O(2,1,R) symmetry, requires λ<0.

A Schwarzschild‐like interior solution for charged spheres
View Description Hide DescriptionA Schwarzschild‐like interior solution for charged spheres is presented in this paper. The solution is regular everywhere.

Jordan–Kaluza–Klein type unified theories of gauge and gravity fields
View Description Hide DescriptionWe investigate the generalized Jordan–Kaluza–Klein type scalar tensortheories of gravity with gauge fields present for the purpose of restricting the spacetime dependence of the scalar fields. These scalars are essential in building Lagrangians for fields with internal degrees of freedom. By one rather simple consistency restriction on covariant differentiation we are able to show that the scalar fields must be spacetime constants.