Volume 35, Issue 12, December 1994
Index of content:

Bound states and inverse scattering for the Schrödinger equation in one dimension
View Description Hide DescriptionThe one‐dimensional Schrödinger equation is considered when the potential and its first moment are absolutely integrable. When the potential has support contained on the left (right) half‐line, it is uniquely constructed by using only the reflection coefficient from the right (left). The bound state norming constants determine whether the potential has support contained on a half‐line or on the full‐line. The bound state energies and the unique set of norming constants yielding the potential with support contained on the left (right) half‐line are completely determined by the reflection coefficient from the right (left). An explicit example is provided.

Modular algebras in geometric quantization
View Description Hide DescriptionPresented here is a discussion on the connection between geometric quantization and algebraic quantization. The former procedure relies on a construction of unitary irreducible representations that starts from co‐adjoint orbits and uses polarizations, while the latter depends on the purely algebraic characterization of unitary irreducible representations, which is based on central decompositions of von Neumann algebras in involutive duality, and their decompositions in terms of maximal Abelian subalgebras. Intermediate stages of these two quantization methods turn out to be complementary, leading thus to a new characterization of the so‐called discrete series representations.

Perturbed factorization of the singular anharmonic‐oscillator eigenequation
View Description Hide DescriptionThe perturbed‐ladder‐operator method is applied to the solution of the perturbed eigenequation {(d ^{2}/dx ^{2})−[m(m+1)/x ^{2}]−b ^{2} x ^{2}+V(x)+Λ}Ψ θix)=0 where V(x)=b _{1}(1/x)^{2}+b _{2}(1/x)^{4}+... is a singular perturbation. This method, which is the extension of the Schrödinger–Infeld–Hull factorization method within the perturbation scheme, provides closed form expressions of the perturbed eigenvalues and ladder functions, by means of algebraic manipulations. As an illustrative application, an analytical solution of the spiked‐harmonic‐oscillator eigenequation {(d ^{2}/dx ^{2})−b ^{2} x ^{2}−(λ/x ^{4})+E}Ψ(x)=0 is worked out up to the second order of the perturbation, by considering specifically adapted m‐ and λ‐dependent perturbing and unperturbed potentials in order to tentatively avoid the known difficulties of convergence of the perturbation series. Closed form expressions of the λ/x ^{4}‐anharmonic‐oscillator energies are obtained in terms of the coupling constant λ and the quantum number v: results following from these expressions are compared with exact available values.

Bifurcation solutions of the Schwinger–Dyson equation
View Description Hide DescriptionAs a function of the coupling constant, it is shown that the Schwinger–Dyson equation has many bifurcation solutions that branch off from the trivial solution. Three of these solutions are obtained explicitly, using the parameter imbedding method and the Liaponov–Schmidt method. Application of these methods to other nonlinear problems is discussed.

W*‐KMS structure from multi‐time Euclidean Green functions
View Description Hide DescriptionAn axiomatic approach to the problem of reconstruction of a dynamics from the Euclidean multi‐time Green functions is proposed. The existence of a W*‐algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown. A condition under which the reconstruction from the one‐time and multi‐time Green functions are equivalent is formulated.

Free fields for any spin in 1+2 dimensions
View Description Hide DescriptionFree fields of arbitrary spin in 1+2 dimensions, i.e., free fields for which the one‐particle Hilbert space carries a projective isometric irreducible representation of the Poincaré group in 1+2 dimensions have been constructed. These representations have been analyzed in detail in the fiber bundle formalism and afterwards Weinberg procedure was applied to construct the free fields. Some comments concerning axiomatic field theory in 1+2 dimensions are also made.

Event structures in nonstandard quantum mechanics
View Description Hide DescriptionEvent structures in nonstandard quantum mechanics are represented by closed subspaces of a nonstandard Hilbert space. There are four natural types of closed subspaces of a nonstandard Hilbert space and these different types coalesce in the standard theory. These types are the closed internal subspaces, the splitting subspaces, the ⊥‐closed subspaces, and the norm‐closed subspaces. The algebraic properties of these four types are studied and compared. Moreover, the resulting structures are compared to others that have recently appeared in the literature.

Stochastic interpretation of Feynman path integral
View Description Hide DescriptionA class of functions H_{ n } ^{F} was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈H_{ n } ^{F} and the initial wave function ψ∈H_{ n } ^{F} then the solution of the Schrödinger equation is ψ_{ t }∈H_{ n } ^{F}. The unitary evolution ψ → ψ_{ t } can be expressed by an integral with respect to the Wiener measure. To each ψ_{ t }∈H_{ n } ^{F} is associated a complex Markov processq _{ t } fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to q _{ t }. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.

The classification of decoherence functionals: An analog of Gleason’s theorem
View Description Hide DescriptionGell‐Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert spaceV; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=tr_{V⊗V}(α⊗βX) for some operator X on the tensor‐product space V⊗V.

A new approach for the Jeffreys–Wentzel–Kramers–Brillouin theory
View Description Hide DescriptionA new approximation to obtain solutions for the time‐independent Schrödinger equation, similar to the classical Jeffreys–Wentzel–Kramers–Brilloin theory, is obtained by means of the theory of continued fractions. A particular case is discussed in detail.

The two‐fermion relativistic wave equations of constraint theory in the Pauli–Schrödinger form
View Description Hide DescriptionThe two‐fermion relativistic wave equations of constraint theory are reduced, after expressing the components of the 4×4 matrix wave function in terms of one of the 2×2 components, to a single equation of the Pauli–Schrödinger type, valid for all sectors of quantum numbers. The potentials that are present belong to the general classes of scalar, pseudoscalar, and vector interactions and are calculable in perturbation theory from Feynman diagrams. In the limit when one of the masses becomes infinite, the equation reduces to the two‐component form of the one‐particle Dirac equation with external static potentials. The Hamiltonian, to order 1/c ^{2}, reproduces most of the known theoretical results obtained by other methods. The gauge invariance of the wave equation is checked, to that order, in the case of QED. The role of the c.m. energy dependence of the relativistic interquark confining potential is emphasized and the structure of the Hamiltonian, to order 1/c ^{2}, corresponding to confining scalar potentials, is displayed.

Optics, mechanics, and quantization of reparametrization‐invariant systems
View Description Hide DescriptionIn this article the dynamics obtained from the Fermat principle is regarded as being the classical theory of light. The action is (first) quantized and it is shown how close one can get to the Maxwelltheory. It is shown that quantum geometricoptics is not a theory of fields in curved space. Considering classical mechanics to be on the same footing, the parallelism between quantum mechanics and quantum geometricoptics is shown. It is shown that, due to the reparametrization‐invariance of the classical theories, the dynamics of the quantum theories is given by a Hamiltonian constraint. Some implications of the above analogy in the quantization of true reparametrization‐invariant theories are discussed.

Equivariance, BRST symmetry, and superspace
View Description Hide DescriptionThe structure of equivariant cohomology in non‐Abelian localization formulas and topological field theories is discussed. Equivariance is formulated in terms of a nilpotent Becchi–Rouet–Stora–Tyutin (BRST) symmetry, and another nilpotent operator which restricts the BRST cohomology onto the equivariant, or basic sector. A superfield formulation is presented and connections to reducible [Batalin–Fradkin–Vilkovisky (BFV)] quantization of topological Yang–Mills theory are discussed.

An algebraic extension of Dirac quantization: Examples
View Description Hide DescriptionAn extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac’s original proposal. These issues play an important role especially in the context of nonlinear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.

Fully relativistic plasma dispersion function
View Description Hide DescriptionProperties of a relativistic plasmadispersion function (PDF) that is required for the description of waves propagating perpendicular to a magnetic field in a fully relativistic, magnetized, Maxwellianplasma, are presented. Series, asymptotic series, recurrence relations, integral representations, derivatives, generating functions, approximations, differential equations, and connections with standard transcendental functions are discussed, as are the PDF’s analytic properties.

Numerical twistor procedure for solving a nonlinear field equation
View Description Hide DescriptionThis paper concentrates on an integrable SU(2) chiral equation in two space and one time dimensions, admitting soliton solutions. It is, in effect, a reduction of the self‐dual Yang–Mills equations in 2+2 dimensions, and can therefore be solved by twistor methods. However, only some solutions can be constructed explicitly, and to solve a general initial‐value problem requires some kind of numerical computation. The paper describes a way of implementing the twistor solution procedure numerically in order to do this.

On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions
View Description Hide DescriptionThe dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n‐) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrödinger equation. A Painlevé singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n‐) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the form f(r)=ε_{1} r ^{−2(n−1)}+ε_{2} r ^{−(n−2)}. This is confirmed by obtaining the associated Lax pair, Bäcklund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one‐dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher‐dimensional inhomogeneous spherically symmetric spin models can be formally mapped.

On a class of ‘‘skewed’’ self‐similar and hyperbolic fractals
View Description Hide DescriptionProbabilistic (recurrent) renewal theory is used herein to describe the geometry of a new class of self‐similar distributions in a ‘‘skewed’’ multiplicative mass splitting procedure. It is shown how this model class can account for a deterministic and conservative treelike structure, in connection with recent developments on ‘‘anomalous’’ multiplicative multifractals. Coexistence of both micro‐ and macroscopic scales in such objects is a consequence of the underlying renewal phenomena being transient, as is shown in a second part of the article. Various ‘‘entropylike’’ measures of the intrinsic disorder prevailing in these structures are proposed.

Closed orbits and their stable symmetries
View Description Hide DescriptionWhat will be shown is the first‐order perturbation of closed orbits for two‐body problems using approximate symmetry theory. A complete group classification with respect to approximate symmetries of the first‐order perturbation of the closed orbit equations corresponding to Kepler’s law and Hooke’s law is made. A discussion then follows of the connection between closed orbits and their stable symmetries.

On the Noether identities for a class of systems with singular Lagrangians
View Description Hide DescriptionA straightforward method for obtaining the generators of Lagrangian gauge transformations within the Lagrangian formalism is presented. This procedure can be carried out completely without the need of developing the Dirac–Hamilton formalism. Singular Lagrangians which may generate Lagrangian constraints are considered. It is assumed that the consistency conditions on the Lagrangian constraints do not generate new independent equations for the accelerations. From the structure of these consistency conditions, the Noether identities are explicitly constructed and the Lagrangian generators of gauge transformations are obtained. It is shown that in the generic case of systems having chains of Lagrangian constraints up to a level k (k≥2), the variations δq under which the action is invariant depend on q, dq/dt, and higher time derivatives up to d ^{ kq }/dt ^{ k }. The relation between this Lagrangian approach and the standard Dirac–Hamiltonian approach is discussed.