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Volume 25, Issue 8, August 1984

The exponential mapping in Clifford algebras
View Description Hide DescriptionThe exponential mapping takes the Lie algebra of the Lorentz group into the Lorentz group. Each element of the group is defined as a formal power series, while the product of two exponential elements usually involves an infinite sum of commutator terms, such as in the Baker–Campbell–Hausdorff formula. Because of the special arithmetic in Clifford algebras, many Baker–Campbell–Hausdorff‐like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra. We calculate exact identities for the Baker–Campbell–Hausdorff formula and related formulas in the quaternion and dihedral algebras. These are useful in treatments of the Lorentz group, and make possible a truly finite (as opposed to infinitesimal) description of a transformation group in physics.

A construction relating Clifford algebras and Cayley–Dickson algebras
View Description Hide DescriptionA review of the applications of the octonions in physics is given. A construction is presented. Both the Cayley–Dickson algebras and the Clifford algebras arise naturally under this construction from the quaternion algebras. The mathematical properties of the algebras constructed are discussed.

Partially coherent states of the real symplectic group
View Description Hide DescriptionIn the present paper, we introduce partially coherent states for the positive discrete series irreducible representations 〈λ_{ d }+n/2,...,λ_{1}+n/2〉 of Sp(2d,R), encountered in physical applications. These states are characterized by both continuous and discrete labels. The latter specify the row of the irreducible representation [λ_{1}λ_{2}⋅⋅⋅λ_{ d }] of the maximal compact subgroup U(d), while the former parametrize an element of the factor space Sp(2d,R)/H, where H is the Sp(2d,R) subgroup leaving the [λ_{1}λ_{2}⋅⋅⋅λ_{ d }] representation space invariant. We consider three classes of partially coherent states, respectively, generalizing the Perelomov and Barut–Girardello coherent states, as well as some recently introduced intermediate coherent states. We prove that each family of partially coherent states forms an overcomplete set in the representation space of 〈λ_{ d }+n/2,...,λ_{1}+n/2〉, and study its generating function properties. We show that it leads to a representation of the Sp(2d,R) generators in the form of differential operator matrices. Finally, we relate the latter to a boson representation, namely a generalized Dyson representation in the cases of Perelomov and Barut‐Girardello partially coherent states, and a generalized Holstein–Primakoff representation in that of the intermediate partially coherent states.

The Kostant partition function for simple Lie algebras
View Description Hide DescriptionA simple algorithm is developed for evaluating the Kostant partition function for any simple Lie algebra. The algorithm may also be used to express the partition function for one Lie algebra in terms of the partition function for another, with the latter algebra not necessarily a subalgebra of the former. A special role in the algorithm is played by the sum of the simple roots. Explicit, closed‐form expressions are given for the partition function for a variety of special cases.

Shape functions for separable solutions to cross‐field diffusion problems
View Description Hide DescriptionThe shape function S(x), which arises in the study of nonlinear diffusion for cross‐field diffusion in plasmas, satisfies the equation S″(x)+λa(x)S ^{α}(x)=0, 0<x<1, S(0)=S(1)=0, α>0. In the cases of physical interest a(x) possesses an integrable singularity at some point in (0,1) but is otherwise continuous. Existence of a positive solution to this problem is established.

A new method for summation of divergent power series
View Description Hide DescriptionA new method for summation of divergent power series is developed. It only requires the knowledge of the form of both the small and large λ‐power expansion (λ being the perturbation parameter) and few coefficients of one of them to yield excellent results. Convergence is proved for a simple two‐level model, and reasonable arguments are given for more complex and interesting models. The method is quite general and contains some resummation techniques reported previously as particular cases. The anharmonic, mean square, displacement function, the ground‐state eigenvalue of the quantum‐mechanical anharmonic oscillator, and the ground‐state energy of the hydrogen atom in a magnetic field calculated in this way are shown to be of striking accuracy in the whole range of the perturbation parameter.

Poisson structure of the equations of ideal multispecies fluid electrodynamics
View Description Hide DescriptionThe equations of the two‐ (or multi‐) fluid model of plasma physics are recast in Hamiltonian form, following general methods of symplectic geometry. The dynamical variables are the fields of physical interest, but are noncanonical, so that the Poisson bracket in the theory is not the standard one. However, it is a skew‐symmetric bilinear form which, from the method of derivation, automatically satisfies the Jacobi identity; therefore, this noncanonical structure has all the essential properties of a canonical Poisson bracket.

Probability manifolds
View Description Hide DescriptionThe concept of a probability manifoldM is introduced. The global properties of M inherited from its local structure are then considered. It is shown that a deterministic spin model due to Pitowski falls within this general framework. Finally, we construct a phase‐space model for nonrelativistic quantum mechanics. These two models give the same global description as conventional quantum mechanics. However, they also give a local description which is not possible in conventional quantum mechanics.

Probability distributions with given multivariate marginals
View Description Hide DescriptionA method is presented for obtaining joint probability density functions which satisfy given multivariate marginal densities.

Maximum entropy in the problem of moments
View Description Hide DescriptionThe maximum‐entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem. In important special cases, such as the Hausdorff moment problem, we establish necessary and sufficient conditions for the existence of a maximum‐entropy solution and examine the convergence of the resulting sequence of approximations. A number of explicit illustrations are presented. In addition to some elementary examples, we analyze the maximum‐entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems. We also briefly indicate possible applications to the Lee–Yang theory of Ising models, to the summation of divergent series, and so on. The general conclusion is that maximum entropy provides a valuable approximation scheme, a serious competitor of traditional Padé‐like procedures.

The connection between variational principles in Eulerian and Lagrangian descriptions
View Description Hide DescriptionThe question of whether there exists a connection between variational principles in Eulerian and Lagrangian descriptions is investigated. By having recourse to a proper view of the Eulerian description it is shown that a variational principle in one description holds whenever a corresponding variational principle in the other description is given. This theoretical conclusion is operative in that a precise rule for writing the new Lagrangian is exhibited. As an application, a new Lagrangian for fluid dynamics in the Eulerian description is determined.

A generalization of the Siewert–Burniston method for the determination of zeros of analytic functions
View Description Hide DescriptionThe Siewert–Burniston method for the derivation of closed‐form formulas for the zeros of sectionally analytic functions with a discontinuity interval along the real axis (based on the Riemann–Hilbert boundary value problem in complex analysis) is generalized to apply to the determination of the zeros of analytic functions (without discontinuity intervals) inside or outside simple smooth contours. An application of this method to the closed‐form solution of the transcendental equation z e ^{ z } =b e ^{ b }, appearing in the theory of neutron moderation in nuclear reactors, is also made.

Closing and abelizing operatorial gauge algebra generated by first class constraints
View Description Hide DescriptionA canonical transformation of operators of relativistic phase space is constructed, which abelizes the basis of the operator‐valued gauge algebra of first class constraints. The new operators of contraints commute among themselves. The new Hamiltonian commutes with the constraints. Dynamics of the new operators is physically equivalent to the initial one.

Blow‐up regularization of singular Lagrangians
View Description Hide DescriptionWe propose a method for regularizing singular Lagrangians by adding new degrees of freedom. We illustrate this regularization method with some particular examples. This procedure is shown to be very useful when the Lagrangian is homogeneous of degree one in the velocities giving rise to an identically vanishing Hamiltonian.

Anharmonic oscillators and generalized hypergeometric functions
View Description Hide DescriptionA large class of anharmonic oscillators represented by the Hamiltonian H(q, p)= 1/2 p ^{2}+ 1/2 ω^{2} _{0} q ^{2} +λq ^{α} (α integer >2) is considered. Owing to an integration technique using the Lagrange–Bürmann theorem, one can give for the period and the action integral of bounded motions closed expressions in terms of energy in the form of very simple generalized hypergeometric functions. Finally an application of the method is given for doubly anharmonic oscillators.

Hamilton–Jacobi theory for constrained systems
View Description Hide DescriptionWe extend the Hamilton–Jacobi formulation to constrained dynamical systems. The discussion covers both the case of first‐class constraints alone and that of first‐ and second‐class constraints combined. The Hamilton–Dirac equations are recovered as characteristic of the system of partial differential equations satisfied by the Hamilton–Jacobi function.

The equivalence problem for nonconservative mechanics
View Description Hide DescriptionA geometric study of the equivalence problem for nonconservative mechanical systems is presented. Three equivalence relations for mechanical systems arise naturally in this framework: L‐, G‐, and H‐equivalence. Necessary and sufficient conditions for the H‐equivalence of two systems are derived. The connection between G‐transformations, G‐equivalence, and canonical transformations is investigated. Furthermore, the relationship to (geometric) quantization is discussed.

On a pointlike relativistic massive and spinning particle
View Description Hide DescriptionA pointlike massive and spinning relativistic particle is described as a confined system of two massless directly interacting spinning constituents. The approach is Hamiltonian. The employed phase space is, thus, a symplectic vector space equipped with global canonical and Poincaré‐covariant twistor coordinates. The Poincaré‐invariant generator of the phase space motion does not represent the energy of the total system. Consequently, the evolution parameter cannot be identified with the time. The generating function, however, makes the position four vector and the proper time of the composite massive and spinning system into dynamical variables, i.e., functions of the evolution parameter. The phase flow may thus be interpreted as a simple particle dynamics in Minkowski space. In analogy with the definition of Bakamjian and Thomas for the center of energy of a relativistic massive and spinning particle, a definition of the center of energy of a massless particle with nonvanishing helicity is presented.

Electromagnetic effects in an elastic circular cylindrical waveguide
View Description Hide DescriptionThe problem of wave propagation in an elastic, thin circular cylindrical tube, including the possibility of propagation of rotating waves, is examined with the use of a nonlinear electromagnetic membrane theory. Some related special cases, which include wave propagation in a class of magnetic polarized rigid tubes and rigid plates, are also briefly discussed.

The squared eigenstates of the sine–Gordon eigenvalue problem
View Description Hide DescriptionAn analysis of the squared sine–Gordon eigenvalue problem in laboratory coordinates is presented. It is shown that unlike the unsquared laboratory coordinate eigenvalue problem, the squared laboratory coordinate eigenvalue problem may be cast into the form of a standard eigenvalue problem, wherein an eigenvalue independent operator operating on an eigenfunction generates the eigenvalue. With this form, it becomes rather elementary to obtain the squared‐eigenfunction expansion of the sine–Gordon potentials as well as to demonstrate closure.