Volume 28, Issue 4, April 1987
Index of content:

An angular momentum analysis of symmetric products of paired nucleon states
View Description Hide DescriptionPair correlated angular momentum projected standard Weyl tableau states spanning an m‐dimensional paired shell‐model space have been obtained for a system of 2N identical nucleons. A simple procedure has been developed for carrying out the restriction of the unitary group U(m) to the rotation group O(3) for configurations of both single and multilevel distributions of the single particle states. The mapping of the correlated pair configuration space onto the antisymmetric states allowed by Pauli principle is achieved using the particle antisymmetrizer. The procedure for determining the shell‐model Hamiltonian matrix using the above basis is outlined.

Representations of N=2 extended supergravity and unitarity conditions in Osp (N,4)
View Description Hide DescriptionThe structure of unitary irreducible representations of the Lie superalgebra Osp (2,4), the algebra of N=2 extended supergravity, is investigated in detail. In particular, four new classes of shortened multiplets are found, and the complete unitarity conditions are given. The shortened multiplets are shown to correspond to atypical infinite‐dimensional representations. Finally, unitarity conditions for Osp(N,4), 3≤N≤8, are constructed.

The asymptotic calculation of Wiener integrals occurring in quantum and statistical physics: A divergency problem
View Description Hide DescriptionThe asymptotic expression of a Wiener functional integral using the eigenvalues of the associated Sturm–Liouville equation is obtained, correcting previous formulas giving a divergent result.

Comment on ‘‘Compact expression for Löwdin’s alpha function’’ [J. Math. Phys. 2 6, 940 (1985)]
View Description Hide DescriptionRecently Antone [J. Math. Phys. 2 6, 940 (1985)] has presented a compact expression for Löwdin’s α function, and expected that, in calculation on the electronic properties of molecules and solids, it is the most convenient of all the expressions available in literature. But, in the present paper, it is shown that the expression for the α function obtained from the expansion formula of Silverstone and Moats [Phys. Rev. A 1 6, 1731 (1977)] is remarkably more compact and, therefore, more convenient than Antone’s expression. Though not visibly, it is proved with the aid of some manipulation that the formula of Silverstone and Moats can be reduced to the well‐known expansion formula of a solid harmonic. In the special case where the center of the function to be expanded and that of the α function both are located on the z axis, some mistakes are found that lead to Antone’s expression for the α function.

A class of self‐dual solutions for SU(2) gauge fields on Euclidean space
View Description Hide DescriptionBy applying the method of separation of variables a particular class of solutions, depending on the four variables (x _{μ}) of Yang’s equation in the R gauge for self‐dual SU(2) gauge fields on Euclidean four‐dimensional flat space, is obtained. These solutions are parametrized by a particular form of the fifth Painlevé transcendent. For a specific choice of certain constants, there is degeneration, and the solutions functionally depend on elementary functions.

An example of ∂̄ problem arising in a finite difference context: Direct and inverse problem for the discrete analog of the equation ψ_{ x x }+uψ=σψ_{ y }
View Description Hide DescriptionThe direct and inverse spectral problem for the discrete analog of the equation ψ_{ x x }+uψ=σψ_{ y } is solved in the framework of ‘‘∂̄’’ theory. The time evolution of the spectral data for the simplest nonlinear differential‐difference equations associated to this linear problem is derived.

Eigenvalues and eigenvectors of the finite Fourier transform
View Description Hide DescriptionThe eigenvalues and eigenvectors of the n×n unitary matrix of finite Fourier transform whose j, k element is (1/(n)^{1/2})exp[(2πi/n)j k], i=(−1)^{1/2}, is determined. In doing so, a multitude of identities, some of which may be new, are encountered. A conjecture is advanced.

Variational calculus on supermanifolds and invariance properties of superspace field theories
View Description Hide DescriptionThe foundations of a variational calculus on fibered supermanifolds are outlined, giving applications to the formulation of superspace field theories. Utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved.

Classical particles with internal structure. II. Second‐order internal spaces
View Description Hide DescriptionA systematic study of classical relativistic particles with internal structure, initiated in a previous paper, is continued and a study of second‐order internal spaces (SOS) is presented within the framework of the Lagrangian form of constrained dynamics. Such internal spaces Q are those for which a phase‐space treatment must necessarily use the cotangent bundle T*Q. The large variety of possible SOS’s—ten discrete cases and two one‐parameter families—is separated into those capable of a manifestly covariant description, and those for which special methods based on the transitive action of SL(2,C) on a coset space are needed. The concept of the isotopy representation plays an important role in this context. Seven of the possible discrete SOS’s are shown to be describable in a manifestly covariant way; two discrete and one one‐parameter family of SOS’s are shown to be unphysical in the sense that no Lagrangians can be written in which the internal and the space‐time position variables are nontrivially coupled; and the remaining single discrete and one one‐parameter family are shown to be physical though not describable in a manifestly covariant way. General phase‐space methods uniformly applicable to all SOS’s are developed; and as an illustrative example a Lagrangian model for an SOS in which the internal space is spanned by two orthonormal spacelike unit vectors is presented.

Hamiltonization for singular and nonsingular mechanics
View Description Hide DescriptionA Hamiltonization procedure valid for both singular and nonsingular mechanics is proposed. A comparison with Dirac’s theory (for singular systems) is developed.

On the strength of Maxwell’s equations
View Description Hide DescriptionThe ‘‘strength’’ of a set of field equations (first defined by Einstein as the number of Taylor coefficients of field variables that could be chosen arbitrarily) is used to show that the amount of initial data required by the electromagnetic formulation of Maxwell’stheory in free space is equal, without approximation, to that required by the potential formulation. In each formulation, the strength is interpreted in terms of the amount of initial data required to provide a solution of the Cauchy initial‐value problem and in terms of the invariance properties of the formulation. Equality of the strengths of the two formulations of Maxwell’stheory is used to support the assertion that knowledge of the strengths of other established field theories provides a means for predicting the possible existence of unknown formulations of the theories.

The equivariant inverse problem and the Maxwell equations
View Description Hide DescriptionIn affirmative, the equivariant inverse problem for Maxwell‐type Euler–Lagrange expressions is solved. This allows the proof of the uniqueness of the Maxwell equations.

Markovian limit for a reduced operation‐valued stochastic process
View Description Hide DescriptionOperation‐valued stochastic processes give a formalization of the concept of continuous (in time) measurements in quantum mechanics. In this article, a first stage M of a measuring apparatus coupled to the system S is explicitly introduced, and continuous measurement of some observables of M is considered (one can speak of an indirect continuous measurement on S). When the degrees of freedom of the measuring apparatus M are eliminated and the weak coupling limit is taken, it is shown that an operation‐valued stochastic process describing a direct continuous observation of the system S is obtained.

Algebra representations on eigenfunctions of the Rosen–Morse potential
View Description Hide DescriptionThe bound state eigenfunctions of the Rosen–Morse [N. Rosen and P. M. Morse, Phys. Rev. 4 2, 210 (1932)] potential are investigated using ladder operators that give a representation of the algebraA _{1}. The representations are generally infinite‐dimensional and indecomposable, the representation space containing unbounded functions as well as the normalizable eigenfunctions. Operators giving eigenfunctions of a potential with different strength are also found, giving a representation of D _{2}. The A _{1} (or su_{2}) representations are identified in terms of a classification proposed by Sannikov (S. S. Sannikov, Yad. Fiz. 6, 1294 (1967) [Sov. J. Nucl. Phys. 6, 939 (1968)]), and the connection with representations obtained by Gruber and Klimyk [B. Gruber and A. U. Klimyk, J. Math Phys. 1 9, 2009 (1978); 2 5, 755 (1984)] is given. The evaluation of matrix elements is considered.

On the problem of local hidden variables in algebraic quantum mechanics
View Description Hide DescriptionGiven two Bose‐type quasilocal C*‐algebras A, B, their state spaces E(A), E(B), and a positive, unit preserving map L: B→A, respecting the local structure of A and B(A,E(A)) is said to have (B,E(B)) as a local hidden theory via L if for all states φ in E(A), L*φ can be decomposed in E(B) via a subcentral measure into states with pointwise strictly less dispersion than the dispersion of φ. After motivating this definition of local hidden theory it is shown that if, in addition, L factorizes on disjoint local algebras, then (B, E(B)) is not a local hidden theory of (A,E(A)) via L.

Addition theorems for spherical wave solutions of the vector Helmholtz equation
View Description Hide DescriptionAddition theorems for spherical wave solutions of the vector Helmholtz equation are discussed. The theorems allow one to expand a vector spherical wave about a given origin into spherical waves about a shifted origin. A simplified derivation of the results obtained earlier by Cruzan [O. R. Cruzan, Q. Appl. Math. 2 0, 33 (1962)] is presented.

Exact solution of the Schrödinger equation with noncentral parabolic potentials
View Description Hide DescriptionThe Schrödinger equation with a class of parabolic potentials has been studied. A constant of motion related to angular momentum has been calculated. It has been shown that the scattering states for these potentials can be defined only in some special cases. The Hartmann potential and the Aharonov–Bohm potential are studied as special cases.

Commutation relations for creation–annihilation operators associated with the quantum nonlinear Schrödinger equation
View Description Hide DescriptionUsing the method of intertwining operators, commutation relations are rigorously obtained for the creation–annihilation operators associated with the quantum nonlinear Schrödinger equation.

Unitary implementability of gauge transformations for the Dirac operator and the Schwinger term
View Description Hide DescriptionRecent interest in external field problems led to the determination of the classes of unitarily implementable gauge, axial gauge, and chiral transformations for the Dirac operator on a finite interval. Various charge quantization conditions are obtained. The algebra of charge operators is worked out and the well‐known Schwinger term is identified.

Weak rigidity in the PPN formalism
View Description Hide DescriptionThe influence of the concept of weakly rigid almost‐thermodynamic material schemes on the classical deformations is analyzed. The methods of the PPN approximation are considered. In this formalism, the equations that characterize the weak rigidity are expressed. As a consequence of that, an increase of two orders of magnitude in the strain rate tensor is obtained.