Index of content:
Volume 27, Issue 1, January 1986

Structure of spin groups associated with degenerate Clifford algebras
View Description Hide DescriptionClifford algebras over finite‐dimensional vector spaces endowed with degenerate quadratic form contain a nontrivial two‐sided nilpotent ideal (the Jacobson radical) generated by the orthogonal complement of such spaces. Thus, they cannot be faithfully represented by matrix algebras. Following the theory of spin representations of classical Clifford algebras, the left regular (spin) representations of these degenerate algebras can be studied in suitably constructed left ideals. First, structure of the group of units of such algebras is examined for a quadratic form of arbitrary rank. It is shown to be a semidirect product of a group generated by the radical and the group of units of a maximal nondegenerate Clifford subalgebra. Next, in the special case of corank 1, Clifford, pin, and spin groups are defined an their structures are described. As an example, a Galilei–Clifford algebra over the Galilei space‐time is considered. A covering theorem is then proved analogous to the one well known in the theory of spin and orthogonal groups.

Hamiltonian particle mechanics on curved space‐times—A no‐interaction theorem
View Description Hide DescriptionHamiltonian particle mechanics is formulated on a space‐time (M,g). The restrictions imposed by the isometry group G of g on the possible dynamics are analyzed; a no‐interaction theorem is obtained on a large class of homogeneous space‐times. The results are in particular applied to the de Sitter space‐time: here it is proven that a G‐invariant dynamics can describe geodesic particle motion only.

Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles
View Description Hide DescriptionNonlinear ordinary differential equations admitting a superposition principle based on the action of the group SL(n,C) on the homogeneous spaces SL(n,C)/O(n,C) and SL(n,C)/Sp(n,C) are derived. The superposition formulas are presented explicitly. In the O(n,C) case the general solution is expressed in terms of three particular solutions (for any n). For the Sp(n,C) case three solutions are needed for n=2k≥8, four solutions for n=6, and five solutions for n=4.

A class of unitary representations of the Lie group Sp(3, R), its coherent states, and its map to a symplectic realization on sp^{*}(3, R)
View Description Hide DescriptionUnitary representations from the positive discrete series of Sp(3, R) with lowest weight w={w _{0} w _{0} w _{0}} are considered. By use of new relations on the enveloping algebra, the generators are constructed as differential operators acting on functions of six real variables. Coherent states for these representations are constructed with the help of the Iwasawa decomposition and used to map the representation space to a symplectic realization on the dual sp*(3, R).

The metaplectic group within the Heisenberg–Weyl ring
View Description Hide DescriptionThe Heisenberg–Weyl ring contains the metaplectic group of canonical transforms acting unitarily on L ^{2}(R). These ring elements are characterized through (i) the integral transform kernels, (ii) coset distributions, and (iii) classical functions under any quantization scheme. The isomorphism under group composition leads to several new relations involving twisted products and quantization of Gaussian classical functions. The Wigner inversion operator is a special central group element. It is shown that the only quantization scheme invariant under metaplectic transformations is the Weyl scheme. The structure studied here appears to be relevant to the study of wave optics with aberration.

Generating relations for reducing matrices. I. Ordinary representations
View Description Hide DescriptionAuxiliary groups are constructed that make it possible to reduce the multiplicity problem and to derive consistent generating relations for the elements of reducing matrices. Three examples are worked out to illustrate the general scheme.

Crossing rules in Cartesian and standard coordinate systems
View Description Hide DescriptionA vectorial and a tensorial crossing rule are defined in Cartesian coordinate systems. Different applications are given. With a particular choice of the standardization coefficient of vectors the same expression of dot and vector products in the standard and Cartesian coordinate systems results. The crossing rules are thus redefined in a standard coordinate system. Applications are obtained in the simplification of some particular ‘‘3n j’’ coefficients.

Relation between the supertableaux of the supergroups OSP(2‖2) and SU(1‖2)
View Description Hide DescriptionThe classification and the interpretation of the Young supertableaux of the orthosymplectic group OSP (2‖2) are given. A comparison is made with the supertableaux of the superunitary group SU(1‖2) taking advantage of the isomorphism of the corresponding superalgebras.

Occurrence of secular terms in the Carleman embedding
View Description Hide DescriptionSecular terms occur in many perturbative solutions of nonlinear equationsystems. In this work, an investigation is made of which cases they may occur in as the result of the application of the linear Carleman embedding to a system of nonlinear equations. The solution for the embedded system is written in a form that makes it convenient to see how these terms originate. Their occurrence for the general case is discussed and the results are exemplified by working out the Hénon–Heiles system.

On the stability of dense point spectrum for self‐adjoint operators
View Description Hide DescriptionLet A be a (random) self‐adjoint operator with fixed orthonormal eigenvectors, but with independently distributed random eigenvalues. [Typically, for the eigenvalue distributions, A is considered to have a dense point spectrum almost surely (a.s.).] A class of perturbations {B} is exhibited such that A+B has only point spectrum a.s. Examples are also constructed, including a rank‐one perturbation B, such that A+μB has no eigenvalues (for μ≠0) a.s., despite A having dense point spectrum a.s.

Factorization of systems of differential equations
View Description Hide DescriptionIt is shown that the classical Infeld–Hull factorization method can be extended to coupled systems of second‐order equations. A complete solution of the factorization equations in two dimensions is given and a partial enumeration of factorizable systems is made.

Some practical observations on the predictor jump method for solving the Laplace equation
View Description Hide DescriptionThe best conditions for the application of the predictor jump (PJ) method in the solution of the Laplace equation are discussed and some practical considerations for applying this new iterative technique are presented. The PJ method was remarked on in a previous article entitled ‘‘A new way for solving Laplace’s problem (the predictor jump method)’’ [J. M. Vega‐Fernández, J. F. Duque‐Carrillo, and J. J. Peña‐Bernal, J. Math. Phys. 2 6, 416 (1985)].

Tree graphs and the solution to the Hamilton–Jacobi equation
View Description Hide DescriptionA combinatorial method is used to construct solutions of the Hamilton–Jacobi equation. An exact expression for Hamilton’s principal function S is obtained for classical systems of finitely many particles interacting via a certain class of time‐dependent potentials. If x, p, and t are the position, momentum, and time variables for N point particles of mass m, it is shown that Hamiltonians of the form H(x,p,t)=(1/2m)p ^{2}+v(x,t) have complete integrals S that are analytic functions of the inverse mass parameter m ^{−} ^{1} in a punctured disk about the origin. If v(x,t) is bounded, C ^{∞} in the x variable, and has controlled x‐derivative growth, then the coefficients of the Laurent expansion of S about m ^{−} ^{1}=0 may be expressed in terms of gradient structures associated with tree graphs. This series expansion for S(x,t; y,t _{0}) converges absolutely, and uniformly for all x, y for time displacements ‖t−t _{0}‖<T≡2K ^{−} ^{1}(m/e U)^{1} ^{/} ^{2}, where K and U are bounds associated with the space derivatives of the potential. For ‖t−t _{0}‖<T, the classical path (from any initial space‐time configuration y,t _{0} to any final configuration x,t) induced by S is unique, passes through no conjugate points, and furnishes the action functional with a strong minimum. The local solution S given above may be used to obtain the classical trajectories for arbitrarily large times.

Symmetries and integrability of the cylindrical Korteweg–de Vries equation
View Description Hide DescriptionThe direct scheme to test integrability of a given nonlinear equation proposed by Chen, Lee, and Liu is tested on the cylindrical Korteweg–de Vries equation. The explicit dependence on t of this equation does not present any real difficulties. Constants of motion and symmetries are found readily and the Lax operators for the scattering problem constructed accordingly.

Positivity and unimodality as stabilizers of the analytic extrapolation of a function known with errors
View Description Hide DescriptionPositivity and unimodality hypotheses on an unknown function χ_{1}(x) confer Stieltjes character to another functionH _{1}(z), known in a discrete set of real points and affected by errors caused by experimental measurements, and impose constraints on the coefficients of its formal expansion which limit the universe of approximant functions, so acting as stabilizers of the analytic extrapolation. The type I Padé approximants, built with the coefficients of the formal expansion, provide rigorous bounds on the function in the cut complex plane. The application of a Stieltjes–Chebyshev technique allows approximations to the function, even on the cut, to be obtained. The physical problem of K ^{±} p forward elasticscattering is approached by the previous method, and bounds on the coupling constant and real part of the amplitude are found.

Some algebras of unbounded operators in an indefinite inner product vector space
View Description Hide DescriptionA general theory of unbounded representations of *‐algebras in the Krein space is formulated. Following the corresponding theory of unbounded representations in the Hilbert space, the connection between states and representations, properties of covariant representations, the notion of irreducibility, and decomposition into irreducible parts are discussed.

A metric space construction for the boundary of space‐time
View Description Hide DescriptionA distance function between points in space‐time is defined and used to consider the manifold as a topological metric space. The properties of the distance function are investigated: conditions under which the metric and manifoldtopologies agree, the relationship with the causal structure of the space‐time and with the maximum lifetime function of Wald and Yip, and in terms of the space of causal curves. The space‐time is then completed as a topological metric space; the resultant boundary is compared with the causal boundary and is also calculated for some pertinent examples.

Dimensional reduction of invariant linear connections and tensor fields on multidimensional space‐time
View Description Hide DescriptionLet the compact Lie groupG act smoothly on the C ^{∞}manifoldS with a single orbit type G/H, where H is a closed subgroup of G, and let N(H)/H be a Lie group [N(H) denotes the normalizer of H in G]. Assuming a given connection in the fiber bundle S → M, where M=S/G is the orbit space (to be identified with the physical space‐time) and N(H)/H is the structure (gauge) group, the G‐invariant tensor fields and linear connections on S are analyzed. A kind of ‘‘dimensional reduction’’ for these objects is established: every field corresponds uniquely to a field on M, and the linear connection defines uniquely a set of fields and a linear connection on M.

Geometrical properties of the algebraic spinors for R ^{3} ^{,} ^{1}
View Description Hide DescriptionThe different geometrical properties of Majorana, ‘‘even,’’ Dirac, and Chevalley spinors in the Clifford algebra approach are investigated.

On the generation of equivalent Hamiltonians
View Description Hide DescriptionA new approach to the equivalence problem (in phase space) is presented. Given a Hamiltonian describing a system of particles with two degrees of freedom (and the corresponding Hamilton–Jacobi equation), it is shown how to find the most general family of Hamiltonian functions that generates a new Hamilton–Jacobi equation with the following (and essential) characteristic, here defined as equivalence: Every new solution is also a solution of the original Hamilton–Jacobi equation and vice versa.