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Volume 27, Issue 2, February 1986

Deformation and contraction in Clifford algebras
View Description Hide DescriptionSpin ideals of Clifford algebras over quadratic spaces of any rank are constructed through contractive limits of suitably deformed classical spin ideals of nondegenerate Clifford algebras. The deformation of the algebraic structures, including the standard Witt basis for the quadratic space, results only from a deformation of the underlying quadratic form. It is shown that a contractive limit of deformed twistor spaces, considered as spin ideals of the Dirac–Clifford algebra, provides a decomposable representation space for the Galilei–Clifford algebra. The limit spin ideals of degenerate Clifford algebras are then decomposed into indecomposable Clifford modules.

Coherent states of the real symplectic group in a complex analytic parametrization. I. Unitary‐operator coherent states
View Description Hide DescriptionIn the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λ_{ d }+n/2,...,λ_{1}+n/2〉 encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the unitary‐operator coherent states, as defined by Klauder, Perelomov, and Gilmore. These states are parametrized by the points of the coset space Sp(2d,R)/H, where H is the stability group of the Sp(2d,R) irreducible representation lowest weight state, chosen as the reference state, and depends upon the relative values of λ_{1},...,λ_{ d }, subject to the conditions λ_{1}≥λ_{2}≥ ⋅ ⋅ ⋅ ≥λ_{ d }≥0. A parametrization of Sp(2d,R)/H corresponding to a factorization of the latter into a product of coset spaces Sp(2d,R)/U(d) and U(d)/H is chosen. The overlap of two coherent states is calculated, the action of the Sp(2d,R) generators on the coherent states is determined, and the explicit form of the unity resolution relation satisfied by the coherent states in the representation space of the irreducible representation is obtained. The Hilbert space of analytic functions arising from the coherent state representation is studied in detail. Finally, some applications of the formalism developed in the present paper are outlined. In particular, its relevance to the study of boson realizations of the Sp(2d,R) algebra is stressed.

The basis for the symmetric irreducible representations of SO(9)
View Description Hide DescriptionAn explicit basis is constructed for the symmetric irreducible representation (irrep) of SU(9)⊇SO(9)⊇SO(5)×SU_{1}(2)×SU_{2}(2). It is also indicated how good angular momentum states can be constructed. The techniques used are based on the well‐known tensor algebra for the infinitesimal generators of simple Lie groups.

Remarks on bifurcation with symmetry, gradient property, and reducible representations
View Description Hide DescriptionThe gradient property for bifurcation equations covariant with respect to a group representation that is reducible, but irreducible as a real representation, is examined. In this case, the Schur lemma does not hold in the usual form, and one is faced with problems not present in the irreducible case. It is shown how to handle these problems, and applications to the fundamental real representations of SO(2) and SU(2) are presented that, due to general group theoretical results, represent in a sense the more general situation.

A unified treatment of SU(N) inner and outer multiplicities
View Description Hide DescriptionA combinatorial approach is developed for calculation of weight multiplicities at or near the center of a weight diagram. The result may be used to determine the inner multiplicity of such a weight, or to decompose the product of several representations into irreducible summands.

A simple construction of twist‐eating solutions
View Description Hide DescriptionA simple general construction of all solutions to the set of equations [Ω_{μ}, Ω_{ν}]=exp (2πi n _{μν}/N)I, where Ω_{μ}∈SU(N) or U(N) and μ, ν=1, 2, ... , 2g, is given.

Differentiation of retarded integrals and the divergence theorem for retarded functions with discontinuities
View Description Hide DescriptionTheorems expressing the time derivatives of retarded volume and surface integrals are presented as well as the Gauss divergence theorem for retarded functions with discontinuities. These theorems greatly facilitate the analysis of gravitational radiation from the motion of disjoint matter distributions in general relativity and could find useful application in other branches of physics.

Transmission through a system of potential barriers. II. Necessary condition for complete transparency. A maximum transmission problem
View Description Hide DescriptionA smooth one‐dimensional system of N potential barriers of arbitrary shapes (unequal or equal) is considered. A general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system. A maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition. The sub‐ and superbarrier cases are treated in a unified way. The exact final formulas can readily be converted into accurate approximate ones by insertion of available phase‐integral expressions (of an arbitrary order) for certain characteristic quantities appearing in the formulas.

Another identity among squares of eigenfunctions
View Description Hide DescriptionA variant of an identity of H. P. McKean and E. Trubowitz [Commun. Pure Appl. Math. 2 9, 143 (1976)] for Hill’s equation is derived via contour integration. The identity is 1=∑^{∞} _{ j=0}(−1)^{ j }‖y _{2}(1, λ_{i}j) ‖ ⋅ f ^{2} _{ j }(x).

Fermions in the space‐time R×S ^{3}
View Description Hide DescriptionQuantum field theories on the surface of a four‐dimensional sphere are considered. The Hamiltonian is rotation invariant and its eigenvalues are discrete. Scalar, vector, and spinorial functions on S ^{3} are discussed. The most general Lagrangians for Dirac, Weyl, and Majorana fermions are derived. They are different from the ones in existing literature. The wave functions and propagator are obtained and formulas for matrix elements involving spinors are presented. The discrete symmetries—parity, charge conjugation, and time reversal—are described. The Lagrangian in R×S ^{3} transforms in a nontrivial way under these. Finally, the fermionic Lagrangian is rederived using the tetrad formalism, and conformal transformations are discussed. This leads to a generalization of the formalism to a time‐dependent radius of curvature. As a particular case, a new Lagrangian for de Sitter space is obtained, which, however, is not invariant under the full de Sitter group.

On the classical limit of phase‐space formulation of quantum mechanics: Entropy
View Description Hide DescriptionThe classical limits of phase‐space formulation of quantum mechanics are studied. As a special example, some properties of both quantum mechanical and classical entropies are discussed in detail.

Numerical integration in many dimensions. III
View Description Hide DescriptionExtending a previous line of work, a powerful computational method is found for numerical integration in many dimensions of functions of the form F ( f _{1}(x _{1} ,x _{2})+f _{2}(x _{2} ,x _{3})+ f _{3}(x _{3} ,x _{4}) +⋅⋅⋅+f _{ d }(x _{ d },x _{1})).

On the Poisson brackets of differentiable generators in classical field theory
View Description Hide DescriptionThe canonical formulation of field theory on open spaces is considered. It is proved, under appropriate assumptions, that the Poisson bracket of two differentiable generators is also a differentiable generator.

Remarks on the Yang–Mills equations in four dimensions: CP^{2}
View Description Hide DescriptionThis paper examines properties of solutions to the Yang–Mills equations in four dimensions and in particular on the manifold CP^{2}: Two solutions are found: one is neither self‐dual nor anti‐self‐dual but is a solution of the full Yang–Mills equations, the other is a self‐dual solution.

Hamiltonian and non‐Hamiltonian perturbation theory for nearly periodic motion
View Description Hide DescriptionKruskal’s asymptotic theory of nearly period motion [M. Kruskal, J. Math. Phys. 4, 806 (1962)] (with applications to nonlinear oscillators, guiding center motion, etc.) is generalized and modified. A new more natural recursive formula, with considerable advantages in applications, determining the averaging transformations and the drift equations is derived. Also almost quasiperiodic motion is considered. For a Hamiltonian system, a manifestly Hamiltonian extension of Kruskal’s theory is given by means of the phase‐space Lagrangian formulation of Hamiltonian mechanics. By performing an averaging transformation on the phase‐space Lagrangian for the system (L → L̄) and adding a total derivative d S/dτ, a nonoscillatory Lagrangian Λ=L̄+d S/dτ is obtained. The drift equations and the adiabatic invariant are now obtained from Λ. By truncating Λ to some finite order in the small parameter ε, manifestly Hamiltonian approximating systems are obtained. The utility of the method for treating the guiding‐center motion is demonstrated in a separate paper.

Stability of forced nonlinear oscillators via Poincaré map
View Description Hide DescriptionThe behavior of nonlinear oscillatorsx(t) driven by a periodic external force is completely determined by the corresponding Poincaré map, which loses stability only in certain well‐known ways. These translate into different classes of perturbations ξ(t) of x(t) that must be considered. By choosing simple representatives in each class, the stability of approximate solutions can be studied analytically. The Duffing equation is considered as an example. An extra island of stability is predicted for a range of driving forces and this is confirmed by numerical computation.

Relativistic brachistochrone
View Description Hide DescriptionThe trajectory joining two points a _{1} and a _{2}, which minimizes the transit time for a particle, initially at rest, to fall in a uniform gravitational field from a _{1} to a _{2}, is called the brachistochrone. Johann Bernoulli was the first to find an analytical form for the brachistochrone; in 1696, he discovered that the trajectory is a cycloid. In this paper the relativistic generalization of this classic problem is presented. Four separate curves are actually identified: a particle falling in both a uniform electric and uniform gravitational field is considered. The curves that minimize the times of flightmeasured by an observer in a laboratory in which a _{1} and a _{2} are fixed and also the curves that minimize the proper times of flight are found.

Front form and point form formulation of predictive relativistic mechanics. Noninteraction theorems
View Description Hide DescriptionThe front form and the point form of dynamics are studied in the framework of predictive relativistic mechanics. The noninteraction theorem is proved when a Poincaré‐invariant Hamiltonian formulation with canonical position coordinates is required.

Realization of Poincaré group induced by a second‐order ordinary differential system. Noninteraction theorem
View Description Hide DescriptionA generalization of the predictive relativistic mechanics is studied where the initial conditions are taken on a general hypersurface of M ^{4}. The induced realizations of the Poincaré group are obtained. The same procedure is used for the Galileo group. Noninteraction theorems are derived for both groups.

Boson realization from quantum constraints
View Description Hide DescriptionGeneralized Holstein–Primakoff realizations deduced by Deenen, Quesne, and Papanicolaou are obtained directly from the algebraic identities satisfied in collective subspaces by the infinitesimal generators of the corresponding dynamical groups.