Volume 36, Issue 7, July 1995
Index of content:

Canonical coherent states for the relativistic harmonic oscillator
View Description Hide DescriptionIn this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given SL(2,R) representation, once a change of variables z∈C→z _{ D }∈ unit disk is performed. Also introduced are higher‐order, relativistic creation and annihilation operators, â,â^{°}, with canonical commutation relation [â, â^{°}]=1 rather than the covariant one [ẑ, ẑ^{°}]≊ energy and naturally associated with the SL(2,R) group. The canonical (relativistic) coherent states are then defined as eigenstates of â. Finally, a canonical, minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator.

Kicked dynamics on Hilbert spaces: A nonstandard approach
View Description Hide DescriptionThe main purpose of this work is to provide a rigorous interpretation of kicked models, with the aid of nonstandard analysis. These modes are represented by evolution equations involving Dirac’s delta functions of time and the method used consists of approximating them by smooth functions on the *R.

Ray representations of N(≤2)+1‐dimensional Galilean group
View Description Hide DescriptionFor a quantum mechanical system, modeled via unitary or antiunitary representations of a symmetry group G with Lie algebrag, the Hamiltonian H is of special interest. If H is an element of g or of the universal enveloping algebra U(g), a generic time dependence for any element in U(g) is given through the Heisenberg picture. As an example we consider a system with Gal(N) as symmetry group with H as one of the generators. For N≥3 one gets from ray representations the free Schrödinger equation. For N=1 a peculiarity occurs: the ‘‘free’’ equation has an interaction term, which results from the construction and parametrization of unitary ray representations. For N=2 there is another special feature: there exist ray representations of the universal covering group Gal(2)■, which induce no ray representations of Gal(2). Furthermore, for these representations it is not possible to construct a Schrödinger equation.

Operator formalism on the Z _{ n } symmetric algebraic curves
View Description Hide DescriptionOn Z _{ n } symmetric algebraic curves of any genus the Hilbert space of analytic free fields with integer spin is constructed. As an application, an operator formalism for the b–c systems is developed. The physical states are expressed in terms of creation and annihilation operators as in the complex plane and the correlation functions are evaluated exploiting simple normal ordering rules. The formalism is very suitable for performing explicit calculations on Riemann surfaces and, moreover, it gives some insight into the nature of two‐dimensional field theories on a manifold. It is proven, in fact, that the b–c systems on a Z _{ n } symmetric algebraic curve are equivalent to a conformal field theory on the complex plane having as primary operators twist fields and free ghosts. Some consequences of the interplay between topology and statistics are also discussed.

Coherent states, path integral, and semiclassical approximation
View Description Hide DescriptionUsing the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e ^{−iTĤ }, where Ĥ is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.

Topological Yang–Mills theory from non‐commutative geometry
View Description Hide DescriptionIt will be shown that the topological Yang–Mills theory of Witten can be induced from the usual Yang–Mills theory of the second rank antisymmetric tensor field when the matrix derivative of non‐commutative geometry proposed by Connes is incorporated in the superconnection framework. It is done by identifying consistently the antisymmetric tensor field B with the usual field strength F by B=−F.

On Feynman’s approach to the foundations of gauge theory
View Description Hide DescriptionIn 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwelltheory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman’s derivation to a Lorentz covariant form with scalar evolution parameters, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura’s derivation does not lead to the usual Maxwelltheory either, because the force equation depends on a fifth (scalar) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass‐shell constraint which this invariance should guarantee. Tanimura’s derivation is examined in the framework of the proper time method in relativistic mechanics, and the technique of Hojman and Shepley is used to study the unconstrained commutation relations. It is shown that Tanimura’s result then corresponds to the five‐dimensional electromagnetictheory previously derived from a Stueckelberg‐type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman’s program of deriving the Maxwelltheory from commutation relations; the Maxwelltheory emerges as the ‘‘correlation limit’’ of a more general gauge theory, in which it is properly contained.

BRST quantization of gauge theories like SL(2,R) on inner product spaces
View Description Hide DescriptionSome general formulas are derived for the solutions of a BRST quantization on inner product spaces of finite dimensional bosonic gauge theories invariant under arbitrary Lie groups. A detailed analysis is then performed of SL(2,R) invariant models and some possible geometries of the Lagrange multipliers are derived together with explicit results for a class of SL(2,R) models. Gauge models invariant under a nonunimodular gauge group are also studied in some detail.

Some rigorous results for Yang–Mills theories on a cylinder
View Description Hide DescriptionUsing an approach based on the canonical formalism, the Yang–Mills theories on a cylinder are rigorously analyzed. In this way the moduli space A/G, can be explicitly described with A being the space of connections and G the group of gauge transformations. In particular A/G_{0}, G_{0} being the group of the pointed gauge transformations, is diffeomorphic to the structure group of the theory G, whereas A/G is G modulo the group of inner automorphisms. It is also proven that A → G is a principal fiber bundle with structure group G_{0}.

Quantum integrable systems related with symmetric spaces of the groups U(1,2) and Sp(1,2) and Green functions on these spaces
View Description Hide DescriptionSome quantum integrable systems related with the groups U(1,2) and Sp(1,2) are considered. The motion of particles in the potentials g _{1}/sinh^{2} α+g _{2}/cosh^{2} α and g _{1} ^{’}/sin^{2} θ+g _{2} ^{’}/cos^{2} θ is related with the free motion in symmetric spaces of these groups. The integral representations for the Green functions of a free particle on these spaces are given.

Exact solution of the Dirac equation for a Coulomb and scalar potential in the presence of an Aharonov–Bohm and a magnetic monopole fields
View Description Hide DescriptionIn the present paper the problem of a relativistic Dirac electron is analyzed in the presence of a combination of a Coulomb field, a 1/r scalar potential, as well as a Diracmagnetic monopole and an Aharonov–Bohm potential. Using the algebraic method of separation of variables, the Dirac equation expressed in the local rotating diagonal gauge is completely separated in spherical coordinates, and exact solutions are obtained. The energy spectrum is computed and its dependence on the intensity of the Aharonov–Bohm and the magnetic monopole strengths is analyzed.

Effects of an internal angular momentum on the rotation of a symmetrical top
View Description Hide DescriptionThe existence of an internal angular momentum induces nutations and periodic deviations from a mean precession. Motions are classified into three cases. In Case I, the nutation is regular during precessions so that the motion is a wobbling, the top behaves triaxially. This triaxiality may be involved in the triaxial deformations of nuclear shapes in nuclear physics. Case II is a limiting case of Case I at an infinite period of nutation. In Case III, the body symmetry axis is over nutated to cross over and to oscillate around the invariable plane. It is an overnutated wobbling. These three cases can be determined by whether the ratio of the internal angular momentum to the total angular momentum is less than or greater than a critical value.

Topology change and the propagation of massless fields
View Description Hide DescriptionThe massless wave equation on a class of two‐dimensional manifolds consisting of an arbitrary number of topological cylinders connected to one or more topological spheres are analyzed herein. Such manifolds are endowed with a degenerate (nonglobally hyperbolic) metric. Attention is drawn to the topological constraints on solutions describing monochromatic modes on both compact and noncompact manifolds. Energy and momentum currents are constructed and a new global sum rule discussed. The results offer a rigorous background for the formulation of a field theory of topologically induced particle production.

On the definition of the center of mass for a system of relativistic particles
View Description Hide DescriptionThe notion of center of mass is reviewed and three natural definitions for the relativistic regime are proposed. This construction can be explicitly calculated by means of an algorithm which is described below.

Recursive recovery of a family of Markov transition probabilities from boundary value data
View Description Hide DescriptionRecent developments in optical imaging inspired the model of photon transport discussed below. (Infrared radiation is used to image relatively soft and homogeneous tissue.) The difficulty of solving Maxwell’sequations, or even linear transport equations, led to this ‘‘diffuse tomographic’’ model. A recursive scheme for solving the two‐dimensional problem is sketched and the first recursive step is detailed.

Debye potentials and Hamiltonians for magnetic field line flow
View Description Hide DescriptionIt is shown that any divergence‐free vector field invariant under a group of volume‐preserving transformations can be expressed locally in terms of two scalar potentials which depend on two variables only. It is also shown that the corresponding field line equations can be written in Hamiltonian form with one of these potentials as the Hamiltonian.

Affine Lie algebraic origin of constrained KP hierarchies
View Description Hide DescriptionAn affine sl(n+1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non‐linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity‐Bäcklund transformations and interpolate between GNLS and multi‐boson KP‐Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy.

Inverse scattering problem for anisotropic media
View Description Hide DescriptionAn inverse scattering problem for a second order matrix differential equation on the line related to the wave propagation in anisotropic media is studied herein. A reconstruction procedure is given based on the Riemann–Hilbert problem of analytic factorization of matrix functions and a uniqueness theorem is proven.

Modified Korteweg–de Vries equation with generalized functions as initial values
View Description Hide DescriptionIn this article the existence of generalized solutions to the modified Korteweg–de Vries equation u _{ t }−6σu ^{2} u _{ x }+u _{ xxx }=0 is studied. The solutions are found in certain algebras of new generalized functions containing spaces of distributions.

The Riemann–Liouville integral and continuous KP hierarchy
View Description Hide DescriptionExtension of the standard construction of the Kadomtsev–Petviashvili (KP) hierarchy by the use of Riemann–Liouville integral is given. In consequence we obtain the new classes of integer as well as fractional graded KP hierarchies, which are further investigated. The fractional calculus leads to the new generalization of the w _{∞}algebra.