Volume 24, Issue 10, October 1983
Index of content:

Character generators for compact semisimple Lie groups
View Description Hide DescriptionA new character formula is presented which leads automatically to character generators in the positive form. The character generator for B _{3} is constructed as an example.

Character generators for elements of finite order in simple Lie groups A _{1}, A _{2}, A _{3}, B _{2}, and G _{2}
View Description Hide DescriptionElements of finite order in the compact simple Lie groups SU(2), SU(3), SU(4), Sp(4) or O(5), and G(2) are considered. We provide the characters of the elements on irreducible representations of Lie groups by assigning appro‐ priate numerical values to the variables on which the characters of representations of the Lie group depend. In this way we specialize generating functions for the characters of the representations of the Lie groups to the generating functions for characters of the elements of finite order. Particular attention is paid to rational elements, all of whose characters are integers; they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read. Gaussian elements are also studied in detail. Their characters are complex valued with integer real and imaginary parts.

Special properties of the irreducible representations of the proper Lorentz group
View Description Hide DescriptionIt is shown that the finite‐ and infinite‐dimensional irreducible representations ( j _{0}, c) of the proper Lorentz group SO(3,1) may be classified into the two categories, namely, the c o m p l e x‐o r t h o g o n a l and the s y m p l e c t i c representations; while all the integral‐j _{0} representations are equivalent to complex‐orthogonal ones, the remaining representations for which j _{0} is a half‐odd integer are symplectic in nature. This implies in particular that all the representations belonging to the c o m p l e m e n t a r y s e r i e s and the subclass of integral‐j _{0} representations belonging to the p r i n c i p a l s e r i e s are equivalent to r e a l‐o r t h o g o n a l representations. The rest of the principal series of representations for which j _{0} is a half‐odd integer are symplectic in addition to being unitary and this in turn implies that the D ^{ j } representation of SO(3) with half‐odd integral j is a subgroup of the unitary symplectic group USp(2 j+1). The infinitesimal operators for the integral‐j _{0} representations are constructed in a suitable basis wherein these are seen to be complex skew‐symmetric in general and real skew‐symmetric in particular for the unitary representations, exhibiting explicitly the aforementioned properties of the integral‐j _{0} representations. Also, by introducing a suitable r e a l basis, the finite‐dimensional ( j _{0}=0, c=n) representations, where n is an integer, are shown to be r e a l‐p s e u d o‐o r t h o g o n a l with the signature ( n(n+1)/2, n(n−1)/2). In any general complex basis, these representations (0, n) are also shown to be p s e u d o‐u n i t a r y with the same signature ( n(n+1)/2, n(n−1)/2). Further it is shown that no other finite‐dimensional irreducible representation of SO(3,1) possesses either of these two special properties.

A class of finite‐dimensional Lie algebras, the Casimir operators of which are not of finite type
View Description Hide DescriptionA class of complex finite‐dimensional Lie algebras is constructed, the center of the universal enveloping algebra of each element of which is not finitely generated. For the construction of these Lie algebras, we use the counterexample of Nagata answering Hilbert’s 14th problem in the negative.

Dirac general covariance and tetrads. I. Clifford and Lie bundles and torsion
View Description Hide DescriptionThe isomorphic map of Clifford and Lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space‐time is used to construct a fully covariant Dirac spinor theory. The Klein–Gordon equation exhibits a natural spin‐torsion coupling of the Einstein–Cartan form, the torsion coming from the tetrad field. The tetrad connection coefficients are explicitly derived in addition to their relationship to the usual Levi–Cività coefficients. Various topological conditions for vanishing torsion are given. The Dirac and adjoint Dirac equations are obtained from a simple Lagrangian and the structure of the adjoint equation is discussed.

Modeling quantum behavior with standard (nonquantum) probability theory
View Description Hide DescriptionThis paper argues that quantum behavior can be modeled using standard probability theory. To show this, such a model is constructed in which the Lagrangians associated with different paths are random. (This random Lagrangian formulation is equivalent to constrained entropy maximization.) We assume that the random error term varies as a harmonic oscillator over time. (We attribute this to certain properties of measuring devices.) The result is a formula which provides a good qualitative description of the n‐slit interference experiment—indeed the formula is quite similar to the formulas of quantum mechanics. Hence standard probability theory models can describe interference effects so that a quantum probability theory is unnecessary.

The geometric foundations of the integrability property of differential equations and physical systems. I. Lie’s ‘‘function groups’’
View Description Hide DescriptionThis series of papers will attempt to discuss in a systematic way when the dynamical differential equations of a physical system have the ‘‘integrability’’ property. This first paper contains two topics: A description of some general properties of ‘‘function groups’’ and the related geometric structures of Poisson‐cosymplectic manifolds; and the Lax representations for differential equations as a sort of ‘‘quantization’’ of Lie’s ‘‘function groups.’’ The general geometric setting of the ‘‘integrability’’ material in terms of the theory of Ehresmann pseudogroups is also described.

Higher‐order special self‐adjoint equations and particle dynamics
View Description Hide DescriptionWe exhibit a remarkable connection between a hierarchy of higher‐order special self‐adjoint ordinary differential equations and the description of motion of a cluster of particles in classical mechanics. The cluster is assumed to consist of equal mass particles all moving in one dimension. In a perturbation schema based on the first‐order equation of motion of the center of mass point, the time evolution of the moments of order m−1 is governed by the solution of a special self‐adjoint equation of order m. A similar connection exists for the moments of a wave packet in quantum mechanics.

Lagrange equations for a spinning gas cloud
View Description Hide DescriptionLagrange equations are derived for a spinning gas cloud. The rotational and vortex velocities are treated as independent variables and their defining equations as equations of constraint. Application of the formalism of Lagrange multipliers to this case of 27 variables and 18 equations of constraint yields nine final equations which are simple in form and contain only variables explicitly included in the kinetic and potential energy.

Classical theories and nonclassical theories as special cases of a more general theory
View Description Hide DescriptionWe analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.

Value preserving quantum measurements: Impossibility theorems and lower bounds for the distortion
View Description Hide DescriptionExtending previous works on the subject we consider the problem of the limitations to ideal quantum measurements arising from the presence of additive conservation laws and we discuss impossibility theorems and derive lower bounds for the deviations from the ideal schemes, with particular reference to the distorting case.

An analytic formula for u(3)‐boson matrix elements
View Description Hide DescriptionThe u(3)‐boson Lie algebra is the liquid limit of the symplectic algebra sp(3, R). An analytic formula is given for the u(3)‐boson matrix elements in irreducible unitary representations corresponding to the sp(3, R) discrete series. The formula also applies to the generators of the mathematically isomorphic, but physically different, interacting boson algebra.

On the binary collision expansion and resummations of it
View Description Hide DescriptionIt is pointed out that when there are only two interaction terms in the Hamiltonian, the binary collision expansion (BCE) for the scattering operator can be resummed into a closed form. When there are more than two interaction terms, one can resum the BCE into a continued‐fraction‐like form by using repeatedly the two interactions resummation. Concise derivations of the BCE are first given, the time‐dependent BCE being obtained in a form applicable to both quantum evolution operators and classical frequency modulated oscillators.

Charged‐particle off‐shell scattering: common structure of all two‐body off‐shell scattering quantities for Coulomb plus rational separable potentials
View Description Hide DescriptionWe establish a common structure of all two‐body off‐shell scattering quantities (expressed in momentum space or coordinate space) associated with Coulomb plus rational separable potentials. We present expressions in so‐called maximal‐reduced closed form, including new formulas: (i) for the off‐shell Jost state for the Coulomb potential, (ii) for the off‐shell Jost function associated with the Coulomb plus Yamaguchi potential, and (iii) for the scattering, regular, and Jost states in coordinate representation for Coulomb plus simple separable potentials for all l.

Nonsingular modified Lippmann–Schwinger equation for two charged particles
View Description Hide DescriptionWe consider the partial waves of the two‐body Lippmann–Schwinger equation for the T matrix in the case of the sum of the Coulomb plus a more rapidly vanishing potential. Using the knowledge of explicit forms of the integrals determining the operation of the Lippmann–Schwinger operator on the singularities that characterize the on‐shell behavior of the Coulomb‐like T matrix, we extract a manifestly nonsingular integral equation from the Lippmann–Schwinger equation. We show that the half‐ and on‐shell inhomogeneous terms and solutions can be made as smooth as the momentum representation partial projection of the non‐Coulomb part of the potential.

Theory of light cone cuts of null infinity
View Description Hide DescriptionLight‐cone cuts of null infinity are defined to be the intersection of the light cone of an interior point x ^{ a } with the future null boundary of the space‐time, i.e., I^{+}. It is shown how from the knowledge of the set of light‐cone cuts of I^{+}, the interior (conformal) metric can be reconstructed. Furthermore, a differential equation defined only on I^{+} is proposed so that (1) the solution space (the parameters defining the set of solutions) is identified with or defines the space‐time itself and (2) the solutions themselves yield the light‐cone cuts which in turn give metrics conformally equivalent to vacuum solutions of the Einstein equations.

Light cone cuts of null infinity in Schwarzschild geometry
View Description Hide DescriptionLight cone cuts of future null infinity in Schwarzschild geometry are studied here. The future null cone from an arbitrary apex in the space‐time has been constructed, and its intersection with I ^{+} is obtained. Knowledge of the cuts yields a great deal of information about the interior of the space‐time. In particular, we use it to reconstruct the Schwarzschild metric up to a conformal factor.

All null orbit type D electrovac solutions with cosmological constant
View Description Hide DescriptionAll null orbit type D solutions of the Einstein–Maxwell equations with λ are obtained. There are only two families of solutions depending upon whether the complex expansion of the electromagnetic eigenvectors aligned along the double DP vectors is different or equal to zero; they are the null orbit solution with complex expansion, and the five‐parameter free of complex expansion null orbit solution, respectively.

Cluster expansion of the distribution functions for a ground state Fermi system
View Description Hide DescriptionThe spin‐averaged Slater sum of the fermion system is expanded in terms of the square of the ground statewavefunction of a boson system and the ‘‘antisymmetry’’ Ursell function. This expansion is used to obtain the cluster series for the radial distribution function of the fermion system in terms of (−C^{(n)}/S), where C^{(n)} is sum of chains of (−f/S) and (−f h ^{ B } _{2}/S) bonds. The series is further expressed in a more compact form using a function L ^{(n)} defined by Eq. (55), and the ‘‘modified’’ FHNC approximation for the radial distribution function is presented.

Derivation of the generalized Langevin equation by a method of recurrence relations
View Description Hide DescriptionThe generalized Langevin equation was first derived by Mori using the Gram‐Schmidt orthogonalization process. This equation can also be derived by a method of recurrence relations. For a physical space commonly used in statistical mechanics, the recurrence relations are simple and they lead directly to the Langevin equation. The Langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation.