Volume 21, Issue 8, August 1980
Index of content:

Shift operator techniques for the classification of multipole‐phonon states: IV. Properties of shift operators in the G _{2} group
View Description Hide DescriptionA previously developed shift operator method is applied to the G _{2} group, which plays an important role in the classification of nuclear octupole‐phonon states. Expressions which connect quadratic products of the considered shift operators with G _{2} invariants are derived.

Shift operator techniques for the classification of multipole‐phonon states. V. Properties of shift operators in the R(7) group
View Description Hide DescriptionWith a view to obtaining additional label‐generating operators for the classification of octupole‐phonon states a set of shift operators O ^{ k } _{ l } (k=0, ±1, ±2, ±3) in the R(7) group is constructed. Expressions which connect quadratic products of these shift operators are given, and it turns out that besides R(7) invariants the expressions also involve the scalar G _{2} shift operator P ^{0} _{ l } previously studied. The opportunity to arrive at an orthogonal solution of the state labelling problem is discussed.

Generalized Bessel functions and the representation theory of U(2) σ C^{2×2}
View Description Hide DescriptionWe construct the matrix elements of both finite transformations and infinitesimal generators in irreducible representations of the motion group U(2) σ C^{2×2} with the aid of the contraction limit of the analogous structures of U(4). The matrix elements of finite transformations are found to have a structure similar to that of the classical Bessel function in that they contain two inverse gamma matrices which couple Wigner D functions. An integral representation is established and related to the matrix‐valued Bessel functions of Gross and Kunze. By means of the representation property of the matrix elements we obtain a new sum rule for classical Bessel functions and an analog of the binomial theorem for the sum of two 2×2 matrices which involves the U(2) gamma matrix instead of the classical gamma function.

Selection rules for type II Shubnikov space groups
View Description Hide DescriptionClebsch–Gordan coefficients for type II Shubnikov space groups, containing the inversion as point group operation are expressed by simple formulas in terms of convenient Clebsch–Gordan coefficients for the unitary subgroup.

Selection rules for P n3′n
View Description Hide DescriptionClebsch–Gordan coefficients for the type II Shubnikov space group P n3′n are calculated in terms of such coefficients for the unitary subgroup P n3n.

Congruence number, a generalization of SU(3) triality
View Description Hide DescriptionCongruence classes of finite‐dimensional representations of semisimple Lie groups are defined. Each class is characterized by a congruence number. For the group SU(3) the concept reduces to the familiar triality number.

Simple derivation of the Newton–Wigner position operator
View Description Hide DescriptionThe kind of operator algebra familiar in ordinary quantum mechanics is used to show formally that in an irreducible unitary representation of the Poincaré group for positive mass, the Newton–Wigner position operator is the only Hermitian operator with commuting components that transforms as a position operator should for translations, rotations, and time reversal and does not behave in a singular way that contradicts what can be learned from Lorentz transformations in the nonrelativistic limit.

On six‐dimensional canonical realizations of the so(4,2) algebra
View Description Hide DescriptionOn each six‐dimensional symplectic manifold a coordinate‐free realization of the so(4,2) algebra can be constructed, the generators of which satisfy the polynomial relations fulfilled by the so(4,2) generators associated with the Kepler problem. This realization contains as a particular case several realizations of so(4,2) known in the literature. An expression of the symplectic form on a six‐dimensional symplectic manifold, in terms of the so(4,2) generators defined on this manifold, is obtained. In particular, on the six‐dimensional orbit of the SO(4,2) group in so(4,2) this symplectic form coincides with the symplectic form introduced by Kirillov Kostant and Souriau. The symplectic form is given a Darboux expression with the aid of three pairs of canonically conjugated variables, which are a generalization of the Delaunay elements defined in the Kepler problem.

Symmetries of differential equations. II
View Description Hide DescriptionThe importance of the non‐pointlike transformations of symmetry is vindicated in relation with the first integrals. A new first integral of a broad class of systems of second order differential equations is obtained out of a symmetry of them w i t h o u t having to impose the restriction that the system be equivalent to a Lagrangian system. The existence of a reciprocal relationship among the local infinitesimal symmetries (l.i.s.) of a Newtonian system of differential equations and the pseudosymmetries of the associated dynamical system is proved. Several applications are developed, and some open problems concerning the dynamical systems of constant divergence are proposed.

A variational principle for resonances
View Description Hide DescriptionA stationary variational principle for calculation of the complex poles of Green functions is given.

WKB method for systems of integral equations
View Description Hide DescriptionThe WKB theory for vector systems of integral equations is developed herein. A variational technique is used to derive the equations for the WKB amplitudes in x‐space or its dual k‐space. Compact, explicit solutions are obtained in one dimension. When a solution breaks down at a turning point, the dual‐space representation can be used to derive the connection formulas between WKB solutions. These connection formulas are equivalent to the rules of the Furry method. The Furry method is used to show how general golbal‐dispersion relations can be constructed.

Partial inner product spaces. IV. Topological considerations
View Description Hide DescriptionWhereas the third paper in this series dealt with the algebraic structure of partial inner product (PIP) spaces, the present one explores systematically their topological properties. A slightly more restricted object is introduced, that we call an indexed PIP‐space: it consists of a PIP‐space together with a distinguished family of assaying subspaces. The upshot of the analysis is the characterization of two types of indexed PIP‐spaces, called type (B) and type (H), respectively, as the most likely candidates for practical applications; they are simply lattices of Banach, resp. Hilbert spaces. Operators on indexed PIP‐spaces are discussed and conditions are given that guarantee that the domain of any such operator is a vector subspace. Finally, we examine the question of the existence of a central Hilbert space, in the case of a positive definite partial inner product.

An integral transform related to quantization
View Description Hide DescriptionWe study in some detail the correspondence between a function f on phase space and the matrix elements. (Q _{ f })(a,b) of its quantized Q _{ f } between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform:Q _{ f }(a,b) = F{a,b ‖ v}f(v) d v, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Q _{ f } between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.

Nature of superspace
View Description Hide DescriptionIt is demonstrated that the superspace of the supersymmetrytheory can be identified in a natural manner with a family of concrete spinor structures over space–time.

On geometrical properties of spinor structure
View Description Hide DescriptionThe Crumeyrolle group e for four‐dimensional space‐time E is explicitely calculated. It is shown that the complexification of the Lie algebra of the group e is a spinor space. In this manner the condition of the existence of a spinor structure over E, formulated as the reduction of the structure group of the bundle of orthonormal frames to e, enables us to associate the spinor space to each point of space‐time in a continuous way.

Stochastic fields from stochastic mechanics
View Description Hide DescriptionStochastic field theory for a real scalar field, considering both zero and positive temperatures, is developed from complements to Nelson’s stochastic mechanics. These complements include path integral formulas for the moments of the stochastic process, a functional differential equation for the generating functional, and a virial theorem. Using these and Yasue’s nonstandard analysis formulation of stochastic field theory, a rigorous meaning is given to the path integral formulas for the field moments and to the functional differential equation of the field’s generating functional.

On a remarkable class of two‐dimensional random walks
View Description Hide DescriptionThis study describes some properties of random walks in a plane which differ from free random walks through an extra weightfactor (−1) for every crossing of some branchline T. The statistical distribution of these walks is derived, asymptotically for very long walks, in the case that T consists of a half‐line. It is pointed out that these walks are relevant to (1) self‐avoiding random walks in a plane; (2) the simple entanglement problem in polymer physics.

Nth‐order multifrequency coherence functions: A functional path integral approach. II
View Description Hide DescriptionNth‐order multifrequency coherence functions arising in beam propagation through focusing media with random‐axis misalignments and focusing media with additive statistical fluctuations are computed. The analysis is carried out by means of a simple formula which yields exact algorithmic solutions to a class of canonical path integrals.

A simple approximate determination of stochastic transition for the standard mapping
View Description Hide DescriptionThe standard mapping results from a study of nonlinear forced oscillations of a gas in a closed tube. On interpreting stochastic transition as the breaking of all waves during a round trip in the tube, an estimate for the critical value of the relevant parameter is simply derived. The estimate agrees very well with the result of a more elaborate analysis of Greene.

Poincaré–Cartan integral invariant and canonical transformations for singular Lagrangians
View Description Hide DescriptionIn this work we develop the canonical formalism for constrained systems with a finite number of degrees of freedom by making use of the Poincaré–Cartan integral invariant method. A set of variables suitable for the reduction to the physical ones can be obtained by means of a canonical transformation. From the invariance of the Poincaré–Cartan integral under canonical transformations we get the form of the equations of motion for the physical variables of the system.