Volume 38, Issue 5, May 1997
Index of content:

Coherent states in quaternionic quantum mechanics
View Description Hide DescriptionWe develop Perelomov’s coherent states formalism to include the case of a quaternionic Hilbert space. We find that, because of the closure requirement, an attempted quaternionic generalization of the special nilpotent or Weyl group reduces to the normal complex case. For the case of the compact group SU(2), however, coherent states can be formulated using the quaternionic halfinteger spin matrices of Finkelstein, Jauch, and Speiser, giving a nontrivial quaternionic analog of coherent states.

The spectrum of the quartic oscillator
View Description Hide DescriptionThe spectrum of the quartic oscillator defined by the Hamiltonian is calculated. The eigenvalues result from a fourthorder linear difference equation by means of recurrence relations. For the symmetric oscillator various eigenvalues have been compared with values obtained from other approaches. In addition, results for the asymmetric oscillator are presented.

The isotropic oscillator
View Description Hide DescriptionWe have obtained in configuration space the state functions of the isotropic oscillator by laddering the state functions of the linear oscillator. Starting from the Hermite functions of the linear oscillator one obtains the associated Laguerre functions of the isotropic oscillator. The energy spectrum is obtained from the curvature of the wave function at a point where the potential energy vanishes.

The covariant oscillators and qdeformed quantum mechanics in n dimensions
View Description Hide DescriptionIn this paper the coherent state for covariant oscillators is constructed and is shown to satisfy the completeness relation. And the qanalog of quantum mechanics in n dimensions is obtained by using oscillators.

Hypergeometric states and their nonclassical properties
View Description Hide Description“Hypergeometric states,” which are a oneparameter generalization of binomial states of the singlemode quantized radiation field, are introduced and their nonclassical properties are investigated. Their limits to the binomial states and to the coherent and number states are studied. The ladder operator formulation of the hypergeometric states is found and the algebra involved turns out to be a oneparameter deformation of algebra. These states exhibit highly nonclassical properties, like subPoissonian character, antibunching, and squeezing effects. The quasiprobability distributions in phase space, namely the and the Wigner functions are studied in detail. These remarkable properties seem to suggest that the hypergeometric states deserve further attention from theoretical and applicational sides of quantum optics.

Pseudoclassical model for Weyl particle in 10 dimensions
View Description Hide DescriptionA pseudoclassical model to describe Weyl particle in 10 dimensions is proposed. In the course of quantization both the massless Dirac equation and the Weyl condition are reproduced automatically. The construction can be relevant to Ramond–Neveu–Schwarz strings where the Weyl reduction in the Ramond sector has to be made by hand.

Renormalized path integral in quantum mechanics
View Description Hide DescriptionWe obtain direct, finite, formulations of a renormalized quantum mechanical system with no reference to ultraviolet cutoffs and running coupling constants, in both the Hamiltonian and path integral pictures. The path integral description requires a modification to the Wiener measure on continuous paths that describes an unusual diffusion process wherein colliding particles occasionally stick together for a random interval of time before going their separate ways.

Incorporation of anomalous magnetic moments in the twobody relativistic wave equations of constraint theory
View Description Hide DescriptionUsing a Diracmatrix substitution rule, applied to the electric charge, the anomalous magnetic moments of fermions are incorporated in local form in the twobody relativistic wave equations of constraint theory. The structure of the resulting potential is entirely determined, up to magnetic type form factors, from that of the initial potential describing the mutual interaction in the absence of anomalous magnetic moments. The wave equations are reduced to a single eigenvalueequation in the sectors of pseudoscalar and scalar states The requirement of a smooth introduction of the anomalous magnetic moments imposes restrictions on the behavior of the form factors near the origin, in space. These conditions are satisfied at the oneloop level of perturbation theory. The nonrelativistic limit of the eigenvalueequation is also studied.

Canonically relativistic quantum mechanics: Representations of the unitary semidirect Heisenberg group,
View Description Hide DescriptionBorn proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual positiontime and energymomentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group where H(1,3) is the Heisenberg group in four dimensions. This is the counterpart in this theory of the Poincaré group and reduces in the appropriate limits to the expected special relativity and classical Hamiltonian mechanics transformation equations. This group has the Poincaré group as a subgroup and is intrinsically quantum with the position, time, energy, and momentum operators satisfying the Heisenberg algebra. The representations of the algebra are studied and Casimir invariants are computed. Like the Poincaré group, it has a Little Group for a massive rest frame and a null frame. The former is U(3) which clearly contains SU(3) and the latter is Os(2) which contains SU(2)⊗U(1).

Analytic regularization of the Yukawa model at finite temperature
View Description Hide DescriptionWe analyze the oneloop fermionic contribution for the scalar effective potential in the temperaturedependent Yukawa model. In order to regularize the model a mix between dimensional and analytic regularization procedures is used. We find a general expression for the fermionic contribution in arbitrary space–time dimension. It is found that in this contribution is finite.

Quantization of symplectic tori in a real polarization
View Description Hide DescriptionWe apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with halfdensities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov–Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using halfforms in the definition of the Hilbert space; (ii) by choosing a fourfold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.

Noncommutative geometry beyond the standard model
View Description Hide DescriptionA natural extension of the standard model within noncommutative geometry is presented. The geometry determines its Higgs sector. This determination is fuzzy, but precise enough to be incompatible with experiment.

The spectrum of Liouville operators and multiparticle Hamiltonians associated to oneparticle Hamiltonians with singular continuous spectrum
View Description Hide DescriptionWe study the structure of spectrum of the Liouville operator H_{−}=H⊗I−I⊗H and the twoparticle Hamiltonian H_{+}=H⊗I+I⊗H in some model situations when the corresponding oneparticle Hamiltonian H has singular continuous spectrum. A Hamiltonian H with singular continuous spectrum of Hausdorff dimension one is constructed such that the absolutely continuous spectrum of the operators H _{−} and H _{+} is empty. On the other hand, we prove the existence of a Hamiltonian H with singular continuous spectrum of Hausdorff dimension zero such that the operators H _{−} and H _{+} have nonempty absolutely continuous spectrum. Thus the Hausdorff dimension of the support cannot serve as characteristic of the singular measure of a onebody Hamiltonian that determines the spectral type of the corresponding Liouvillians or twobody Hamiltonians.

Information gains expected from separate and joint measurements of identical spin systems: Noninformative Bayesian analyses
View Description Hide DescriptionAn investigator is assumed to have replicas of a spin system, but lacks any specific knowledge pertinent to constructing a density matrix for it. We compute—within a noninformative Bayesian framework—the expected information gains for various measurement schemes, first, separate and, then, joint in nature. In particular, we obtain a result which fully accords with a certain plausibility argument of Peres, while the parallel result in a recent analysis of Massar and Popescu does not.

Asymptotic completeness for Rayleigh scattering
View Description Hide DescriptionWe consider an electron bound by some anharmonic external potential and coupled to the quantized radiation field in the dipole approximation. We prove asymptotic completeness for the photon scattering. This means that an arbitrary initial state has a long time asymptotic, which consists of electron plus radiation field in their coupled ground state and finitely many outgoing photons.

Wave equations and kinematical integrals for complexified Maxwell–Klein–Gordon fields
View Description Hide DescriptionThe dynamics of Maxwell–Klein–Gordon fields in the framework of complex Minkowski space is considered. It is particularly shown that the procedures involved in the actual derivation of the wave equations which control the propagation of photons lead naturally to new structures describing the corresponding sources. The relevant kinematical integrals are readily built up by utilizing a suitable defining twospinor expression for the energymomentum tensor of the complete theory.

Uniform high frequency approximation to scattering from infinite strip
View Description Hide DescriptionA new technique for approximating the high frequency scattering amplitude from flat obstacles is used to solve the problem of acoustic plane wavescattering from an infinite soft strip. This method yields the leading order terms in a uniform high frequency asymptotic expansion for the scattering amplitude which reduce to the results found with the geometrical theory of diffraction in the regions in which that theory is valid. The asymptotic expressions derived here are compared with exact numerical solutions and are found to be accurate for all angles of incidence and reflection.

Dynamics of generalized Poisson and Nambu–Poisson brackets
View Description Hide DescriptionA unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.

Integrability, Stäckel spaces, and rational potentials
View Description Hide DescriptionFor a variety of classical mechanical systems embeddable into flat space with Cartesian coordinates and for which the Hamilton–Jacobi equation can be solved via separation of variables in a particular curvalinear system we answer the following question. When is the separable potential function expressible as a polynomial (or as a rational function) in the defining coordinates Many examples are given.

The source identification problem in electromagnetic theory
View Description Hide DescriptionThe problem of the identification of the electromagnetic source which produces an assigned radiation pattern is illposed: the solution is, in general, not unique and it does not depend continuously on the data. In this paper we treat in detail these two aspects of the problem. First of all we reconsider the radiation problem in the very general setting of the Sobolev spaces in order to make more acceptable, from a physical viewpoint, the conditions which have to be imposed on the electromagnetic sources. Then by the use of the Euclidean character of the Hilbert spaces we decompose the sources into a radiating and a non radiating component. We determine the subspace of the radiating sources and we find the basis spanning this subspace. Particular attention is then devoted to the case of the linear antenna. In this case the solution of the problem is unique but it does not depend continuously on the data. We may, however, implement the problem taking into account a bound on the ohmic losses. This is sufficient to restore the continuity. Finally a method of variational regularization (in the sense of Tikhonov) is discussed in detail.