Volume 22, Issue 9, September 1981
Index of content:

Quasigroup construction and first class constraints
View Description Hide DescriptionA generalization of the Lie group construction is proposed wherein the composition law depends, apart from the parameters of the transformations composed, also on the transformed variables. This construction is met, in particular, on the hypersurfaces specified by the first class constraints in phase space.

A criterion for completeness of Casimir operators
View Description Hide DescriptionWe give a criterion for a set of Casimir operators of any semisimple Lie algebra to be an algebraically independent generating set of the algebra of the Casimir operators. The criterion is applied to supply complete sets for the Casimir operators of A _{ l }, B _{ l }, C _{ l } , and D _{ l } . With the aid of a method to construct some Casimir operators we then furnish complete sets for the Casimir operators of the exceptional Lie algebras using the criterion.

Casimir operators for F _{4}, E _{6}, E _{7}, and E _{8}
View Description Hide DescriptionWe construct explicitly algebraically‐independent generating sets for the Casimir operators and the invariants of the Weyl groups and adjoint groups of the exceptional Lie algebrasF _{4}, E _{6}, E _{7}, and E _{8}.

The analytical properties of the off‐shell T matrix for infinite rank nonlocal separable potentials
View Description Hide DescriptionAnalytical expression and uniform bounds for the lth partial wave off‐shell T matrix are derived for infinite rank separable potentials. It is proved that Fredholm’s alternative can be used to solve the Lipmann–Schwinger equation in some cases of noncompact nonlocal potentials in the strong L _{ p }topology.

On the stability of periodic orbits of two‐dimensional mappings
View Description Hide DescriptionWe present a closed form stability criterion for the periodic orbits of two‐dimensional conservative as well as ’’dissipative’’ mappings which are analogous to the Poincaré maps of dynamical systems. Our stability criterion has a particularly simple form involving a finite, symmetric, nearly tridiagonal determinant. Its derivation is based on an extension of the stability analysis of Hill’s differential equation to difference equations. We apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied ’’standard mapping’’ describing a periodically ’’kicked’’ free rotator. As another example we also obtain explicitly and in closed form the intervals of bounded (and unbounded) solutions of a discrete ’’Schrödinger equation’’ for the Kronig and Penney crystal model.

Geodesic first integrals with explicit path‐parameter dependence in Riemannian space–times
View Description Hide DescriptionIn a Riemannian space V _{ n } general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend e x p l i c i t l y on the path parameter s. It is found that such first integrals must also be polynomials in s. Necessary and sufficient conditions are found for the existence of these first integrals. The existence of many well‐known symmetries such as homothetic motions (scale change), affine collineations, conformal motions, projective collineations, conformal collineations, or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path‐parameter dependence. To illustrate the theory, geodesic first integrals of this type have been calculated for four Riemannian space–times of general relativity.

Global definition of nonlinear sigma model and some consequences
View Description Hide DescriptionThe (nonlinear) sigma model is defined as a field theory whose configurations are sections of a nontrivial fiber bundle over space–time. The action functional is a generalization of the ’’energy’’ used in the theory of harmonic maps. This definition requires minimal coupling to a Yang–Mills field, and the solutions of the coupled equations exhibit spontaneous symmetry breaking. It is shown that in a Higgs phenomenon making use of a sigma model instead of the Higgs fields, no scalars would survive symmetry breaking.

On the inverse problem of the calculus of variations
View Description Hide DescriptionWe consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first‐order form. We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach. We present examples of first‐order Lagrangians for systems for which no second‐order Lagrangians exist. The Hamiltonian theory for first‐order (degenerate) Lagrangians is constructed using Dirac’s method for singular Lagrangians.

Dynamical Noether invariants for time‐dependent nonlinear systems
View Description Hide DescriptionDynamical invariants are derived for time‐dependent systems with nonlinear equations of motion including nonharmonic damped systems. The concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated. Finally we show the existence of dynamical invariants for some nonlinear quantum systems.

Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media
View Description Hide DescriptionFor the dispersion of waves in a homogeneous medium there exist the Kramers–Kronig relations for the wave number K(ω) = ω/c(ω). The usual mathematical proof of such relations depends on assumptions for the asymptotic behavior of c(ω) at high frequency, which for electromagnetic waves in dielectrics can be evaluated from the microphysical properties of the medium. In this paper such assumptions are removed and the necessary asymptotic behavior is shown to follow the representation of K(ω) as a Herglotz function. From the linear, causal, and passive properties of the media we thus establish the Kramers–Kronig relations for all linear wave disturbances including acoustic, elastic, and electromagnetic waves in inhomogeneous as well as homogeneous media without any reference to the microphysical structure of the medium.

Electromagnetism and the holomorphic properties of spacetime
View Description Hide DescriptionThe Cauchy–Riemann equations of holomorphy are extended to fields in higher‐dimensional spaces in a framework of Clifford algebras. The equations of holomorphy in Minkowski spacetime turn out to be the Maxwell equations in vacuum. The Lorentz gauge condition is a result of the holomorphy. Sources can be included in an extension of the residue theorem, where charges correspond to the residues.

A Hamiltonian structure of the interacting gravitational and matter fields
View Description Hide DescriptionWe present a Hamiltonian formulation for classical field theories. In a general case we write the Hamilton equation by means of the energy–momentum function E and the symplectic 2‐form Ω. We investigate thoroughly an important example, the gravitational field coupled to a matter tensor field. It will be shown that the energy‐momentum differential 3‐form yields a generalization of the Komar energy formula. We prove that the energy–momentum function E, the symplectic 2‐form Ω, the Hamilton equation, and four constraint equations for initial values of canonical variables give rise to the system which is equivalent to the Euler–Lagrange variational equations. We also discuss relations between the Hamilton equation of evolution, the degeneracy of the symplectic 2‐form Ω, and the action of the diffeomorphism group of spacetime in the set of solutions.

Path integration for time‐dependent metrics
View Description Hide DescriptionWe define path integrals for systems with time‐dependent metrics in terms of prodistributions and discuss the relation of the path integral quantization to the Schrödinger one. We work an example that displays the elegance and utility of the prodistribution definition. We also discuss how our definition is particularly suited for changing integration variables.

Dynamical symmetries of the Schrödinger equation
View Description Hide DescriptionThe conditions are derived under which a symmetry operator quadratic in the momenta exists for a spin‐zero structureless point charge interacting with an externally applied uniform magnetic field in the presence of a potential field. The conditions apply to the possible forms of the potential. The explicit form of the symmetry operator in general is constructed and some particular examples are examined.

Exponential perturbations of the harmonic oscillator
View Description Hide DescriptionThe operators H(β) = p ^{2}+x ^{2}+βe ^{ r x }(r∈R{0}) in L ^{2}(R) are studied. The spectrum is discrete for ‖argβ‖<π, the eigenvalues admit an asymptotic expansion as ‖β‖→0, and they have no Bender‐Wu type singularities in the analytic continuation to any punctured sector of a logarithmic Riemann surface. For β′<0,H(β′) defines a symmetric operator with deficiency indices (1,1) and all its self‐adjoint extensions have discrete spectrum; however, any eigenvalue of H(β′), when continued to β′<0, can be interpreted as a resonance of the problem.

Symmetry of time‐dependent Schrödinger equations. I. A classification of time‐dependent potentials by their maximal kinematical algebras
View Description Hide DescriptionPotentials for the time‐dependent Schrödinger equation [− 1/2 ∂_{ x x } +V(x,t)]Ψ(x,t) = i∂_{ t }Ψ(x,t) are classified according to their space–time or kinematical algebras in a search for exactly solvable time‐dependent models. In addition, it is shown that their dynamical algebras are isomorphic to their kinematical algebras on the solution space of the Schrödinger equation.

Partition combinatorics and multiparticle scattering theory
View Description Hide DescriptionThe recently developed combinatoric methods for handling partition‐labeled operators in N‐particle scattering theory are studied from an abstract point of view. The relation of these methods to approaches of the cluster/cumulant type in many areas of mathematical physics is pointed out. The concept of connectedness is defined abstractly and the mathematical structure of the partition lattice is considered in detail. Many of the useful results of combinatoricscattering theory are shown to be natural expressions of properties of the partition lattice. The conditions on these results can then be stated with precision. A number of new operator theorems are also obtained by means of applying simple extensions and analogs of the known properties of the partition lattice.

Eigenvalues of an anharmonic oscillator
View Description Hide DescriptionThe five‐term WKBJ approximation is applied to calculate the eigenvalues for the potential V(x) = 1/2 k x ^{2}+a x ^{4}, k≳0 and a≳0. Numerical results are compared with those of Hioe and Montroll. It is found that the accuracy of the calculated eigenvalues improves rapidly with increase in the quantum number n. At n = 4, a seven significant figure accuracy is achieved and at n = 6, a nine significant figure accuracy.

A nonmanifold theory of space–time
View Description Hide DescriptionA Zeemantopology is defined in the general framework of any set W of events which has been equipped with an acyclic signal relation ∼ →. The Wssumption that the ∼ → structure of W is locally that of Minkowski space and that the ’’piecing together’’ maps are smooth in an appropriate sense, allows a tangent bundle p:E→W to be defined. This bundle has, as structure group, the group G of linear causal automorphisms of Minkowski space.

Local transverse‐traceless tensor operators in general relativity
View Description Hide DescriptionTwo flat‐space transverse‐traceless tensor operators can be used to construct initial data for numerical solutions of the gravitational fieldequations. One of these operators is related to the conformal curvature 3‐tensor and is shown to exist in a large class of nonflat 3‐spaces. The second operator enjoys no such liberty. Important applications to gravitational wave scattering are suggested. It is argued that the number of operators available on a particular 3‐space is related to the number of gravitational field modes that are excited in the space.