Volume 23, Issue 2, February 1982
Index of content:

The unitary irreducible representations of SL(2, R) in all subgroup reductions
View Description Hide DescriptionWe use the canonical transform realization of SL(2, R) in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group. Explicit results are given for all mixed bases and subgroup reductions. These provide the full multiparameter set of integral transforms and series expansions associated to SL(2, R).

Geometric quantization and representations of semisimple Lie groups: spin (2,1) and spin (2,2)
View Description Hide DescriptionThe UIR’s of spin (2,1) and spin(2,2) are studied in the light of Kostant–Auslander induction scheme, and compared with those obtained earlier by Harish–Chandra and Schmid.

The Riccati equation and Hamiltonian systems
View Description Hide DescriptionThe involutive system of functionals associated by Gel’fand and Dikii to a nth‐order scalar differential operator is obtained from a set of solutions of a generalized Riccati equation. These solutions allow us to explain the involutive character of the system of functionals in terms of the Riccati equation properties only.

Factorization of operators and completely integrable Hamiltonian systems
View Description Hide DescriptionGeneralized Miura transformations induced by factorization of an nth‐order scalar operator are used to characterize a set of Hamiltonian systems by requiring the conservation of the Gel’fand–Dikii first integrals sequence. The second symplectic structure for the Gel’fand–Dikii equations is obtained in connection with the previous Hamiltonian systems. Bäcklund transformations are also analyzed.

A global isometry approach to accelerating observers in flat space–time
View Description Hide DescriptionA global two‐point diffeomorphic extension of Lorentz transformations is constructed which preserves the global Lorentzian metric structure of flat R ^{4}. This global mapping induces, as a tangent‐space mapping, instantaneous Lorentz transformations parametrized by interframe velocity functions. The elimination of pseudoterms from particle and electromagnetic field equations leads to an exact analytic expression for the affine connection needed for covariant differentiation. Examination of invariant particle equations gives an obvious proof of the equivalence principle in terms of the symmetric part of the acceleration‐group connection. Transformation properties of the connection coefficients are shown to be in accord with general covariance requirements. The specific case of the rotating observer is treated exactly where it is seen that the affine connection merely supplies the exact Thomas precession term. Recent work by DeFacio e t a l. is found to be especially convenient for comparison with the present work. The results of the two approaches agree precisely. A summary of results indicates that the global isometry approach gives results consistent with those obtained via presymmetry arguments.

Clifford algebra approach to twistors
View Description Hide DescriptionLocal particle interpretation or, equivalently, an enlargement of a structure group to the Poincaré group at each point of a Riemannian space‐time manifold naturally results in a complexification of the Clifford algebra for the tangent Minkowski space. Following Crumeyrolle, twistor space is identified with an appropriate one‐sided ideal of this algebra. Every antiautomorphism of the latter provides a unique projection from the complexified Clifford algebra onto the affine complex Minkowski space. This projection commutes with the action of the Poincaré group. Using the above approach, three projections (the cases of symmetric, antisymmetric, and Hermitian tensors) are derived. The projection in terms of the antisymmetric, decomposable tensors is shown to give the Penrose projection.

Explicit evaluation of certain Gaussian functional integrals arising in problems of statistical physics
View Description Hide DescriptionAn explicit formula is presented for a (conditional) Wiener integral, the integrand of which is an exponential of a general quadratic functional of the path. The functional integrals arising in non‐Markovian Gaussian approximations to various problems of statistical physics (e.g., theory of the large polaron,theory of disordered systems) are easily recovered as special cases.

Random walks on lattices. The problem of visits to a set of points revisited
View Description Hide DescriptionA general method is outlined for calculating the statistical properties of the number of visits to a set of points in a random walk. In illustrative examples, known results and new results are easily derived.

Hypervirial calculation of integrals involving Bessel functions
View Description Hide DescriptionA general and simple procedure is presented for evaluating matrix elements that involve Bessel functions. The method is based upon hypervirial relationships for systems subjected to Dirichlet boundary conditions.

Poincaré–Cartan integral invariant and canonical transformations for singular Lagrangians: An addendum
View Description Hide DescriptionThe results of a previous work, concerning a method for performing the canonical formalism for constrained systems, are extended when the canonical transformation proposed in that paper is explicitly time dependent.

Solution of a Schrödinger inverse scattering problem with a polynomial spectral dependence in the potential
View Description Hide DescriptionThe inverse scattering problem for the scalar Schrödinger equationy″+[E−J_{ p = 0} ^{ n } (E ^{1/2n })^{ p } u _{ p }(x)]y = 0, x∈R, is considered. It is solved by reduction to the inverse scattering problem for a matrix Schrödinger equation:Y″+[E I−(U(x)+E ^{ 1/2 } Q(x))] Y = 0, x∈R.

Generalized second‐order Coulomb phase shift functions
View Description Hide DescriptionSome specific properties and the evaluation of the generalized second‐order Coulomb phase shift functions (two‐dimensional integrals of four spherical cylinder functions) are discussed. The dependence on the three momenta k _{1},k̄,k _{2}, corresponding to the final, intermediate, and initial states is illustrated.

Pre‐atlas manifolds and coordinate‐free general relativity
View Description Hide DescriptionEinstein’s equations can be expressed in the tetrad form so that coordinates do not appear explicitly. Tetrads, however, are usually defined on a manifold, which means that coordinates have been introduced. The notion of a manifold without coordinates (a preatlas manifold) is described here and it is shown that Einstein’s equations can be expressed in this setting without introducing coordinates at any stage. Conditions on a preatlas manifold are given which ensure that a C ^{0} atlas can be generated. The motivation for this formulation is the desire to incorporate the philosophy of relativity, which asserts that the mathematical laws of nature are essentially independent of observers or coordinates. ’’The introduction of numbers as coordinates...is an act of violence.’’—H. Weyl

Structure of asymptotic twistor space
View Description Hide DescriptionWe show that asymptotic projective twistor space PT^{+} is an Einstein–Kähler manifold of positive curvature. We then use the Chern–Moser theory of hypersurfaces in complex manifolds to show that the Kähler curvature of PT^{+} closely related to the CR curvature of its boundary. We also give a proof that the Kähler potential function defining the boundary satisfies the complex Monge–Ampere equations.

Timelike infinity
View Description Hide DescriptionA new formulation is presented for analyzing the structure of a space–time at timelike infinity. An asymptotically simple space–time is defined as a space–time (M,g) which can be imbedded in a space (M̄,ĝ) with boundary T,a C ^{∞} metric ĝ and a C ^{∞}scalar field Ω, such that Ω = 0 on T, Ω≳0 on M−Ta n dĝ^{μν}−ĝ^{μλ}ĝ^{νρ}Ω_{‖λ} Ω_{‖ρ} = Ω^{−2} g ^{μν}−Ω^{−4} g ^{μλ} g ^{νρ}Ω_{;λ}Ω_{;ρ} in a neighborhood of T. Demanding that T = T^{−}U T^{+}, where each one of T^{−} and T^{+} is isometric to the unit spacelike hyperboloid, and ĝ^{μν} Ω_{‖μ} Ω_{‖ν} = Ω^{−4} g ^{μν} Ω_{;μ} Ω_{;ν} = 1 on T, we have an almost asymptotically flat (at timelike infinity) space–time. The group of asymptotic symmetries of (M,g) at timelike infinity is found to be isomorphic to the Lorentz group. Some properties of the space–time near T are shown.

A unified formulation of timelike, null and spatial infinity
View Description Hide DescriptionA formulation is presented for studying simultaneously the timelike, null, and spatial infinities of space‐times which resemble asymptotically Minkowski’s space‐time. For this a relation of the form f(g,ĝ,Ω) = 0 is determined so that given a space‐time (M,g) a space (M,g) with boundary B can be found with M imbedded in M̂ and ĝ and ΩC ^{∞} fields on M̂. A space‐time (M,g) for which this is possible is called (globally) asymptotically simple. Then the conditions for B to resemble the boundary of Minkowski’s space‐time are determined. Thus the concept of a (globally) almost asymptotically flat space‐time is defined.

Electromagnetic energy tensors and the Lorentz equation of motion for fields with electric and magnetic charge distributions
View Description Hide DescriptionIn special or general relativity the electromagnetic energy tensor is usually taken to be ϑ^{ a b } = (1/4π)(F ^{ a } _{ c } F ^{ b c }−(1/4) g ^{ a b } F _{ c d } F ^{ c d }). This expression may also be used in the generalized theory which allows magnetic as well as electric charge. Rund [J. Math. Phys. 18, 84 and 1312 (1977)] has suggested a new approach to the generalized theory with an alternative form for the energy tensor. We show that in Rund’s theory there are other possible definitions for the energy tensor. However, there is a strong indication that a particular energy tensor gives rise in a definite way to a corresponding Lorentzequation of motion. This equation is derived for each of the energy tensors and it is found that only ϑ^{ a b } gives the Lorentzequation which is usually assumed in the generalized theory. Furthermore, the Lorentzequations arising from the other energy tensors will not give charge quantization.

Coupled translational and rotational diffusion in liquids
View Description Hide DescriptionThe equations for coupled translational and rotational diffusion of asymmetric molecules immersed in a fluid are obtained. The method used begins with the Kramers–Liouville equation and leads to the generalized Smoluchowski equation for diffusion in the presence of potentials. Both external potentials and intermolecular potentials are considered. The contraction of the description from the Kramers–Liouville equation to the Smoluchowski equation is achieved by using a combination of operator calculus and cumulants. Explicit solutions to these equations are obtained for the two‐dimensional case. Comparison of our results with earlier literature is also presented.

Two variable relativistic tensor harmonics
View Description Hide DescriptionThree bases in the Hilbert space of tensor fields on the unit spheres associated with two independent vectors are discussed: the tensor spherical harmonics and the symmetric and unsymmetric tensor helicity harmonics. Under the conditions which we specify they form complete sets of independent Lorentz covariants which may serve the purpose of the analysis of reactions with several particles in the final state.

A 4‐vector generalization of the sine–Gordon equation
View Description Hide DescriptionUsing the differential operators from the Dirac equations, an algebra is developed which leads to 4‐vector functions and the generalization of many scalar functions. Assuming particles to be described by the potential solutions of the generalized sine–Gordon equation (a set of four coupled nonlinear equations), a single soliton is shown to be localized within a light sphere and have intrinsic properties of group velocity, phase velocity, angular momentum, and wave‐particle duality.