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Volume 31, Issue 8, August 1990

Algebras of diffeomorphisms of the N‐torus
View Description Hide DescriptionThe general algebraD(N,V) of infinitesimal diffeomorphisms of the N‐torus (S ^{1})^{ N } involving generators L _{ v,m } depending on a structure vector v∈C^{ N } and a vector‐valued index m∈Z^{ N } is constructed. Several results are proved for this algebra. Special cases of the algebra were previously presented in Ramos–Shrock [E. Ramos and R. E. Shrock, Int. J. Mod. Phys. A 4, 4279 (1989).] The concept of the Q rank of the space V of structure vectors, denoted r _{Q}(V), is defined and certain related ‘‘Cartan matrices’’ are introduced. It is shown that D(N,V), as an ungraded algebra, is simple if and only if r _{Q}(V)=N, i.e., the Q rank of V is maximal. A classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the Cartan matrices. The space of structure vectors is isomorphic to the unique ‘‘Cartan subalgebra.’’ A number of properties concerning the central extensions of these algebras are then proved. For the case dim V=1, r _{Q}(V)=N, it is shown that the central extension is unique. For the case dim V=1,r _{Q}(V)<N, a new and greatly enlarged central extension is constructed involving a central charge f u n c t i o n from Z^{ N−r } to C (essentially equivalent to an infinite number of central charge parameters), rather than a single central charge parameter. Finally, it is proved that for dim V≥2, this algebra has no nontrivial central extension.

The complete root systems of the affine Kac–Moody superalgebras
View Description Hide DescriptionBy using certain special automorphisms, the complete root structures of all the affine Kac–Moody superalgebras, both twisted and untwisted, are derived in detail.

On quantum fields that generate local algebras
View Description Hide DescriptionThe properties of quantum fields generating local nets of von Neumann algebras are discussed. For fields that satisfy certain energy bounds, necessary and sufficient conditions for the existence of such a net and for the existence of product states are given in terms of the vacuum expectation values.

Trivector Fourier transformation and electromagnetic field
View Description Hide DescriptionA new kind of Fourier transformation is proposed for distributions taking values in the Clifford algebra of three‐dimensional space, with the unit imaginary replaced by the unit trivector. The transformation is used to introduce special distributions that describe the free electromagnetic field.

Representations of affine Kac–Moody algebras and the affine scalar product
View Description Hide DescriptionSimple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra, for finding the Weyl orbit of an arbitrary weight, and for determining whether or not any given weight is in any given representation. A simple definition of congruency is given that applies to all affine algebras. The standard indefinite scalar product is generalized; the generalization is used in the procedures.

Nontransititive imprimitivity systems for the Galilei group
View Description Hide DescriptionA class of nontransitive imprimitivity systems and the corresponding projective unitary representations for the inhomogeneous Galilei group are worked out with the Mackey–Varadarajan method of group representations.

Some remarks on the Gupta–Bleuler triplet
View Description Hide DescriptionIndecomposable representations of the Poincaré group associated to infrared singular field theory models are discussed in the framework of the general theory of the Gupta–Bleuler triplet formulated by Araki. It is shown that the definition of maximal Hilbert space structures, related to the infrared properties of the states of the models, can be exploited to construct representation spaces for the Gupta–Bleuler triplet. The examples of the two‐dimensional massless scalar field and of the electromagnetic field in the Landau gauge are discussed. In particular, in the first example, the relation between the Gupta–Bleuler triplet and the algebraic treatment of the massless scalar field is investigated. In the case of electromagnetism, the structure of the representation of the Poincaré group in the Landau gauge is clarified. The explicit form of the corresponding Gupta–Bleuler triplet for the one‐particle space of the electromagnetic field is exhibited.

All SL(3,R) ladder representations
View Description Hide DescriptionAll unitary and nonunitary ladder (multiplicity free with respect to the ∼(SO)(3) maximal compact subgroup) representations of the double covering group ∼(SL) (3,R) of the SL(3,R) group are cataloged and presented explicitly. The list of nonunitary representations is corrected and completed, and a new insight concerning the Δj=2 ladder representation starting with j= (3)/(2) is obtained.

Symmetries of hyper‐Kähler (or Poisson gauge field) hierarchy
View Description Hide DescriptionSymmetry properties of the space of complex (or formal) hyper‐Kähler metrics are studied in the language of hyper‐Kähler hierarchies. The construction of finite symmetries is analogous to the theory of Riemann–Hilbert transformations, loop group elements now taking values in a (pseudo‐) group of canonical transformations of a simplectic manifold. In spite of their highly nonlinear and involved nature, infinitesimal expressions of these symmetries are shown to have a rather simple form. These infinitesimal transformations are extended to the Plebanski key functions to give rise to a nonlinear realization of a Poisson loop algebra. The Poisson algebrastructure turns out to originate in a contact structure behind a set of symplectic structures inherent in the hyper‐Kähler hierarchy. Possible relations to membrane theory are briefly discussed.

Classification of star and grade‐star representations of C(n+1)
View Description Hide DescriptionThe two types of * and grade‐* representations of the Lie superalgebraC(n+1) are classified. A type (1) irreducible *‐representation is characterized by the single condition (Λ,α_{ s })≥0, Λ and α_{ s } being the highest weight and the odd simple root, respectively, while the type (2) *‐representations are duals of type (1) *‐representations. For n>1, only the identity and vector representations of C(n+1) are shown to be grade ‐*. The superalgebra C(2) proves to be an exception and admits several classes of nontrivial irreducible grade ‐* representations.

Tight frames of compactly supported affine wavelets
View Description Hide DescriptionThis paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 4 1, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψ_{ j k }(x)=2^{ j/2}ψ(2^{ j } x−k) k j,k∈Z} is a tight frame of L ^{2}(R), i.e., for every f∈L ^{2}(R), f=c∑_{ j k } 〈ψ_{ j k }‖f〉ψ_{ j k } for some c>0. This family is parametrized by an algebraic subset V _{ N } of R ^{4N }. Furthermore, for N≥2, a proper algebraic subset W _{ N } of V _{ N } is specified such that all points in V _{ N } outside of W _{ N } yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

Exact and steady‐state solutions to sinusoidally excited, half‐infinite chains of harmonic oscillators with one isotopic defect
View Description Hide DescriptionA half‐infinite chain of spring‐mass oscillators with nearest‐neighbor coupling is excited by a sinusoidal force applied to the mass at the accessible end of the chain. The identical linear springs are massless. Each mass has measurem(m>0) except one, which has measure μm(μ>0). An exact solution is given for the initial‐value problem in which all initial velocities and displacements are zero. Behavior of the solution for large t (time) is examined. When the concept of average power supplied by the source to the chain in steady state is meaningful, expressions for average power are deduced.

Solution of Laplace’s equation in plane single‐connected regions bounded by arbitrary single curves
View Description Hide DescriptionA method is developed to solve Laplace’s equation with Dirichlet’s or Neumann’s conditions in plane, single‐connected regions bounded by arbitrary single curves. It is based on the existence of a conformal transformation that reduces the original problem to another whose solution is known. The main advantage of the method is that it does not require the knowledge of the transformation itself, so it is applicable even when no transformation is available. The solution and its higher‐order derivatives are expressed in terms of explicit quadratures easy to evaluate numerically or even analytically.

Integrability condition and finite‐periodic Toda lattice
View Description Hide DescriptionA further generalization to construct integrable dynamical systems with hierarchy has been developed, based upon a generalization of the zero Nijenhuis tensor condition in a symplectic manifold with double symplectic structures. As an example, the periodic Toda lattice Hamiltonian has been shown to satisfy these conditions. The case of the three‐dimensional Coulomb potential problem has also been analyzed in a similar fashion.

Particle symmetry and antisymmetry in approximately relativistic Lagrangians
View Description Hide DescriptionFor a single‐time approximation of the type discussed by Woodcock and Havas [Phys. Rev. D 6, 3422 (1972)] applied to particle‐asymmetric Poincaré‐invariant variational principles (VPs) of the Fokker type, a method is presented for expressing approximately relativistic Lagrangians (ARLs) to any order in c ^{−1} in a form such that coefficients of functions of the instantaneous three‐separation r _{ i j } are either particle symmetric or antisymmetric. These functions of r _{ i j } are determined solely by the corresponding particle symmetric or antisymmetric parts of the exactly relativistic kernel of the VP describing two‐body interactions of N classical point particles. While the exact kernel involving the particles’ four‐separations and four‐velocities is particle asymmetric, the built‐in static Newtonian limit is particle symmetric. Using this method to reformulate previously published ARLs to order c ^{−3} makes it obvious that a sufficient condition for acceleration‐free ARLs to order c ^{−3} is that the kernel of the exact non‐time‐reversal‐invariant interaction be particle symmetric.

The Goursat problem for the homogeneous wave equation
View Description Hide DescriptionThe Goursat problem for the 3‐D homogeneous wave equation is presented and some methods to solve it are discussed.

Berry’s phase for photons and topology in Maxwell’s theory
View Description Hide DescriptionThe classical nature of Berry’s phase for photons is shown to arise from the intrinsic topological structure of Maxwell’stheory. The phase is developed in the context of fiber‐bundle theory and is discussed in some detail.

Nth (even)‐order minimum uncertainty products
View Description Hide DescriptionThis paper considers the problem of finding the quantum states that minimize the products of the (even) Nth‐order fluctuation of two canonically conjugate operators. The problem is first attacked in an abstract form and an equation derived for the desired states. A consideration of the N=2 case then leads to a connection with the new concept of ‘‘squeezed states’’ of the electromagnetic field, and the usual exact solution involving the coherent state. The concept of ‘‘higher‐order squeezing’’ is touched on to motivate the further discussion. The cases N=4, 6, and 8 are then taken up, and approximate solutions to them found via a new numerical technique. During these calculations, it is noted that only the first few terms in the expansion in number states figure in the solution; this observation is then exploited to find an approximate closed solution to the problem valid for all N.

Algebraic calculation of the Green’s function for the Hartmann potential
View Description Hide DescriptionUsing the so(2,1) Lie algebra and the Baker–Campbell–Hausdorff formulas, the Green’s function for the Hartmann potential is constructed and its bound state energy spectrum is found. Also, this Green’s function is constructed in a coherent state basis and the equivalence of the two descriptions is shown.

The generalized continued fractions and potentials of the Lennard‐Jones type
View Description Hide DescriptionFor a broad class of the strongly singular potentials V(r), which are defined as superpositions of separate power‐law components, the general solution of the corresponding Schrödinger differential equation is constructed as an analog of Mathieu functions. The analogy is supported by the use of the (generalized) continued fractions. The questions of convergence are analyzed in detail.