Index of content:
Volume 24, Issue 11, November 1983

On Z _{ n } graded Baker–Campbell–Hausdorff formulas and coset space parametrizations
View Description Hide DescriptionWe present a generalization of the BCH formula, exp Δ exp Γ=exp(Δ+Γ+ (1/2)[Δ,Γ]+⋅⋅⋅), involving n factors on the right, linear in Δ and to all orders in Γ. The result is applicable to coset geometries involving Z _{ n } graded algebras and superalgebras. In the Z _{2} case expressions are obtained for group transformations in the spaces S ^{ n−1}=O(n)/O(n−1), CP^{ n }=SU(n+1)/U(n), OSp(1/n)/Sp(n), and SU(n/1)/U(n).

A duality consistent phase convention for complex conjugation in SUn
View Description Hide DescriptionWe have argued that a simple phase convention for all involutive mappings can be chosen so the algebra of inner product couplings in SUn is identical to that for outer products in S _{ L }. Pure real phases with a single ±1 entry in any row or column can represent the permutation matrix for transposition in order of the two component irreducible representations (irreps) in a binary coupling, the matrix for association in S _{ L } with respect to the alternating group A _{ L }, the Derome–Sharp matrix, and the 1j m factor. The latter two are required by complex conjugation in SUn. In our previous work, we have proposed specific prescriptions for assigning the phase under transposition and association and have shown them to be consistent with duality. In this work, we propose a duality consistent prescription for assigning the phase of the Derome–Sharp matrix and thus of the 1j m factor which is related to it by association.

Eigenvalues of Casimir operators for the general linear and orthosymplectic Lie superalgebras
View Description Hide DescriptionA Racah basis is introduced for the generators of these matrix superalgebras and explicit formulas are derived for eigenvalues of Casimir operators in terms of the components of the highest weight. The result contains, as special cases, the corresponding expressions for the general linear, orthogonal, and symplectic Lie algebras.

The use of comparison filters in linear filter theory
View Description Hide DescriptionIn the present paper, it is shown how the linear filter equation for a given correlation coefficient can be solved in terms of the solution of the filter equation with a different correlation coefficient. The second filter is called a comparison filter. One obtains an integral equation for the difference of the two filters in terms of the difference of the two correlation functions and the solution of the comparison filter. Thus if the comparison filter is known and its correlation coefficient is close to that of the desired filter, one may regard the comparison filter as being an approximation to it. The difference of the two filters is then small and perturbation expansions or variational principles for the difference may be expected to give better results than if one did not use a comparison filter. The difference in the solutions of the two filter equations may also be regarded as the change (or error) in the filter due to a change (or error) in the correlation coefficient. Our result is obtained by pressing the close analogy of the filter equation to the Gel’fand–Levitan equation of inverse spectral theory. Another result of the use of comparison filters is to show that the filter equation for the difference of filters satisfies a possibly useful grouplike property.

New representations for the spherical tensor gradient and the spherical delta function
View Description Hide DescriptionIn this article, we analyze representations for the product Y^{ m 1 } _{ l 1 }(∇)F ^{ m 2 } _{ l 2 }(r) with Y^{ m } _{ l }(∇) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r. Since both Y^{ m 1 } _{ l 1 }(∇) and F ^{ m 2 } _{ l 2 }(r) are irreducible spherical tensors, we can use angular momentumalgebra for evaluating the product. Accordingly, the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product. Analytical expressions for these radial functions are derived by direct differentiation and with the help of Fourier transforms. Closely related to the spherical tensor gradient Y^{ m } _{ l }(∇) is the spherical delta function δ^{ m } _{ l }(r). We derive new representations for δ^{ m } _{ l } by considering convolution integrals involving B functions. These functions are closely related to the modified Bessel functions and also to the Yukawa potential e ^{−αr }/r. We show that the definition of the B functions can be extended to include a large class of derivatives of the delta functions, where the spherical delta function is just a special case.

A useful formula for evaluating commutators
View Description Hide DescriptionWe present the derivation of a useful formula for evaluating commutators of the form [A, f (B)] and [ f (A),B], where the nested commutators [A,[ A,[⋅⋅⋅[ A[A,B]]⋅⋅⋅]]] and [[[⋅⋅⋅[[A,B],B ]⋅⋅⋅],B ],B] do not vanish in general. The use of this formula is illustrated by a simple example.

Dirac tensor distributions for moving submanifolds of R ^{ n }
View Description Hide DescriptionThis paper considers nonclassical fields (tensor distributions) of the form τδ_{Ω̃}, where δ_{Ω̃} is the Dirac delta function for a moving p‐dimensional submanifold of R ^{ n } (0≤ p≤n). The density τ is a classical (smooth), rank‐ktensor field on R ^{ n+1}. The main result of the paper is the development of formulas for the distributional derivatives of such fields. The derivatives considered are the absolute differential (Levi‐Civita connection), the covariant derivative along a given vector field, the divergence operator, the exterior differential, and the exterior codifferential. The resulting derived fields are shown to reflect the underlying geometry of the submanifold Ω as well as the nature of its motion. In the special case p=n, it is seen that the jump conditions on fields at the boundary of the region Ω arise naturally from the distributional calculus.

Realization of Gaussian random fields
View Description Hide DescriptionThe representation of stationary Gaussian processes in terms of filtered Gaussian white noise is studied. Known results are extended from the finite‐dimensional case to the dimension‐free case; hence, in particular, to Gaussian random fields. In particular, the following result is proved for usual Gaussian processes: Physical realizability is equivalent to realizability.

First‐order equations of motion for classical mechanics
View Description Hide DescriptionA global canonical first‐order equation of motion is derived for any mechanical system obeying Newton’s second law. The existence of a Lagrangian is not assumed, but the properties of the canonical equation are similar to those of the Hamiltonian formulation. The choice of map F from velocity space to phase space is not determined by the condition that the first‐order equation of motion be equivalent to a second‐order equation on configuration space and therefore is left open to be selected on the basis of other considerations. The canonical equation is a covector or 1‐form equation on the Whitney sum T*Q⊕T Q and contains the second‐order equation condition, restriction to the graph of the map F, and Newton’s equation of motion in first‐order form. The last is related to Newton’s second‐order equations by the consistency condition that the motions not lead off graph F in T*Q⊕T Q. The first‐order equation of motion can be projected onto phase space if the map F can be inverted.

Generalized Hamiltonian dynamics. I. Formulation on T*Q⊕T Q
View Description Hide DescriptionThe Dirac–Bergmann generalized Hamiltonian dynamics for a degenerate‐Lagrangian system is formulated on the Whitney sum T*Q⊕T Q of the phase space T*Q and the velocity space T Q over the configuration space Q. The formulation is related to those on T*Q and T Q. Some ambiguities concerning generalized dynamics that have appeared in the literature are clarified.

Generalized Hamiltonian dynamics. II. Gauge transformations
View Description Hide DescriptionGeneralized Hamiltonian dynamics is the finite‐dimensional version of gauge field theory and possesses invariance properties corresponding to gauge invariances. It is argued herein that a proper description of the finite‐dimensional gauge transformations requires a time‐dependent formalism. With this, local and global gauge transformations can be distinguished, and the insight obtained from this distinction clarifies some ambiguities that have appeared in the literature. In particular, the time‐dependent formalism provides a precise statement for Dirac’s conjecture concerning the form of the generalized Hamiltonian on phase space.

Transient fields in dispersive media
View Description Hide DescriptionThe problem addressed in this paper is the determination of transmitted and scattered fields produced by a transient electromagnetic field incident on a three‐dimensional body when the body and the surrounding medium are allowed to be dispersive. Instead of decomposing the pulse into its Fourier components, the solution is carried out in the time domain to take advantage of marching‐in‐time procedures. Maxwell’sequations are suitably modified, and the reduction of the problem to the solution of an integral equation for a single tangential vector field is adapted to dispersive media. A simple conductor and a collisionless plasma are studied as examples.

On the controllability of quantum‐mechanical systems
View Description Hide DescriptionThe systems‐theoretic concept of controllability is elaborated for quantum‐mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert spaceH of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state‐evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain D_{ω} in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite‐dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of D_{ω} with a suitably defined finite‐dimensional submanifold of the unit sphere S_{H} in H. Several soluble examples are presented to illuminate the general theoretical results.

Convergence of the T‐matrix approach in scattering theory. II
View Description Hide DescriptionConvergence of the T‐matrix scheme is proved under more general assumptions than in Ramm [J. Math. Phys. 2 3, 1123–5 (1982)] and for more general boundary conditions. Stability of the numerical scheme towards small perturbations of data and convergence of the expansion coefficients are established. Dependence of the rate of convergence on the choice of basis functions is discussed. Dependence of the quality of expansions in various spherical waves on the shape of the obstacle is discussed.

Positive‐definite self‐dual solutions of Einstein’s field equations
View Description Hide DescriptionWe investigate (anti‐) self‐dual Riemann space‐times for diagonal Bianchi types of class A with positive‐definite metrics. A general algorithm to find self‐dual solutions is presented. Explicit solutions are given for all types of class A.

Solutions of Einstein–Yang–Mills equations with plane symmetry
View Description Hide DescriptionThe Einstein–Yang–Mills system is solved with the assumption of plane symmetry. We present a class of abelian and a class of SO(3) nonabelian solutions of the Yang–Mills equations.

Structure and motion of the Lee–Yang zeros
View Description Hide DescriptionFor an Ising model satisfying the Lee–Yang condition, the zeros of the partition function Z and those of the associated functions Z _{ A } in the space of imaginary magnetic fields at all lattice sites are determined by a single analytic hypersurface. The sense of motion of the zeros of Z as the interactions are varied can be related to the positions of the zeros of the Z _{ A } . Contrary to a plausible conjecture, it is not true that all of the zeros of Z in a uniform field tend towards the point ẑ=1 in the complex fugacity plane as the temperature is lowered, but it is possible that the first zero (that nearest to ẑ=1) has a monotone motion. Various simplicity and intertwining properties of the zeros of Z and Z _{ A } which generalize earlier results are proved by a new argument which makes direct use of the Lee–Yang property.

Generalized two‐dimensional O(3) sigma model
View Description Hide DescriptionWe generalize the two‐dimensional O(3) nonlinear σ‐model while preserving its conformal invariance. The integrability condition of this model encompasses the sine‐Gordon equation in addition to some special cases which are found to be of the same form. The time‐independent solutions exhibit solitonlike behavior.

Inverse scattering variables of the KdV equation from the point of view of Galilean mechanics
View Description Hide DescriptionThe Galilean invariance of the Korteweg–de Vries equation is applied in order to characterize the structure of degrees of freedom displayed by the inverse scattering transform method. It is found that the dynamical systems associated with the discrete scattering data variables admit a description in terms of mass, position, and momentum variables similar to the systems of free Galilean particles. On the other hand, the radiation component of the KdV field associated with the continuous part of the set of scattering data turns out to be described by a new field evolving according to a linear partial differential equation.

Graded tensor calculus
View Description Hide DescriptionWe develop a graded tensorcalculus corresponding to arbitrary abelian groups of degrees and arbitrary commutation factors. The standard basic constructions and definitions, like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented. Moreover, the graded versions of the classical Lie algebras are introduced, and some of their basic properties are described.