Volume 41, Issue 8, August 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Bispectrality for the quantum Ruijsenaars model and its integrable deformation
View Description Hide DescriptionAn elementary construction of the eigenfunctions for the quantum rational Ruijsenaars model with integer coupling parameter is presented. As a byproduct, we establish the bispectral duality between this model and the trigonometric Calogero–Moser model. In particular, this gives a new way for calculating Jack polynomials. We propose also a certain oneparameter deformation of the Ruijsenaars model, proving its integrability and bispectrality. The generalizations related to other root systems and difference operators by Macdonald are considered.

State vector reduction as a shadow of a noncommutative dynamics
View Description Hide DescriptionA model, based on a noncommutative geometry, unifying general relativity and quantum mechanics, is developed. It is shown that the dynamics in this model can be described in terms of oneparameter groups of random operators, and that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum mechanical approximation gives the standard unitary evolution of observables, and in the “space–time limit” it leads to the state vector reduction. The cases of the spin and position operators are discussed.

Exact evolution operator on noncompact group manifolds
View Description Hide DescriptionFree quantum motion on group manifolds is considered. The Hamiltonian is given by the Laplace–Beltrami operator on the group manifold, and the purpose is to get the (Feynman’s) evolution kernel The spectral expansion, which produced a series of the representation characters for in the compact case, does not exist for noncompact group, where the spectrum is not bounded. In this work real analytical groups are investigated, some of which are of interest for physics. An integral representation for is obtained in terms of the Green’s function, i.e., the solution to the Helmholz equation on the group manifold. The alternative series expressions for the evolution operator are reconstructed from the same integral representation, the spectral expansion (when exists) and the sum over classical paths. For noncompact groups, the latter can be interpreted as the (exact) semiclassical approximation, like in the compact case. The explicit form of is obtained for a number of noncompact groups.

Path space measure for the dimensional Dirac equation in momentum space
View Description Hide DescriptionNonstandard analysis is used to construct a measure over paths for the path integral solution to the Dirac equation in dimension. Paths are considered in momentum space, because the Green function in the configuration space contains a derivative of δ function which keeps us from assigning a measure over paths. The solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure, but also as a standard path integral with respect to a standard measure extracted from the nonstandard one. The result is an extension of Gaveau’s work [B. Gaveau, J. Funct. Anal. 58, 310–319 (1984)].

An inverse scattering problem with the Aharonov–Bohm effect
View Description Hide DescriptionA direct and an inverse scattering method is developed for Schrödinger operators with electromagnetic fields in the case of obstacles in order to study the wellknown Aharonov–Bohm effect. In dimension greater or equal to three, we show that the electric potential and the magnetic field are uniquely determined by the Soperator. In the twodimensional case, some obstruction appears based on a quantification of the magnetic flux.

Generalized Chern–Simons form and descent equation
View Description Hide DescriptionWe present the general method to introduce the generalized Chern–Simons form and the descent equation which contain the scalar field in addition to the gauge fields. It is based on the technique in a noncommutative differential geometry (NCG) which extends the fourdimensional Minkowski space to the discrete space such as with two point space However, the resultant equations are not only dependent on NCG, but also are justified by the algebraic rules in the ordinary differential geometry.

Classical and quantum mechanics with timedependent parameters
View Description Hide DescriptionWe show that composite bundles where is the parameter bundle, provide the adequate mathematical description of mechanical systems with timedependent parameters both in classical and quantum mechanics. In particular, the Berry phase phenomenon is described in terms of connections on composite Hilbert space bundles.

Schrödinger’s equation and general relativity
View Description Hide DescriptionSchrödinger’s equation is generalized to a space–time fourmanifold, using standard concepts from differential geometry and operator replacement. This fourthorder equation, which reduces and specializes to the Klein–Gordon equation in the flat space limit, can also be obtained from a variational principle, and must be solved in tandem with the Einstein field equations with suitable stress energy. The propagator, for large momenta, varies like A further attractive feature is that no external currents or stress energies need be imposed: these arise naturally. A generalization to fields with arbitrary spin is proposed. Solving the equation would lead to a determination of the mass, just as energies are found in solving Schrödinger’s equation. Flatspace plane wave solutions consist of the superposition of two independent waves, which can be interpreted as propagating strings.

Timeoptimal control of finite quantum systems
View Description Hide DescriptionWe investigate timeoptimal control of finite quantum systems in the Born approximation. A bang–bang principle is found to follow from a result in [C. A. Akemann and J. Anderson, Mem. Amer. Math. Soc. 458 (1991)]. We also prove existence of timeoptimal controls, characterize when they are unique, and assuming uniqueness, explicitly describe them.

A representation space for minimal coupling
View Description Hide DescriptionSternberg’s construction of the phase space for minimal coupling combines the phase space of a classical mechanical system with a principal fibre bundle with connection (gauge potential) and a Hamiltonian Gspace. Using geometrical quantization data for a classical system and a Hamiltonian Gspace we construct a representation space for the minimal coupling space.

vertex operators, screen currents, and correlation functions at an arbitrary level
View Description Hide DescriptionBosonized qvertex operators related to the fourdimensional evaluation modules of the quantum affine superalgebra are constructed for arbitrary level where is a complex parameter appearing in the fourdimensional evaluation representations. They are intertwiners among the levelα highest weight Fock–Wakimoto modules. Screen currents which commute with the action of up to total differences are presented. Integral formulas for Npoint functions of type I and type II qvertex operators are proposed.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Gaugeinvariant covariant Hamiltonians
View Description Hide DescriptionGiven an Ehresmann connection γ on a fibered manifold a covariant Hamiltonian density is then associated to each Lagrangian density L on Assume E is the bundle of connections of a principal bundle and that L is gauge invariant. Our goal in this paper is to determine conditions on γ under which is also gauge invariant. The general conclusion is that there is no gaugeinvariant Ehresmann connection but there is plenty of such connections providing gaugeinvariant covariant Hamiltonians. The relevant cases of bundles and bundles are discussed in detail.

Higherorder mechanical systems with constraints
View Description Hide DescriptionA general mathematical theory covering higherorder mechanical systems subject to constraints of arbitrary order (i.e., depending on time, positions, velocities, accellerations, and higher derivatives) is presented, including higherorder holonomic systems as a particular case. Within differential geometric setting on higherorder jet bundles, the concept of a mechanical system (not necessarily regular, or Lagrangian) is introduced to be a class of 2forms equivalent with a dynamical form. Dynamics are then represented by means of corresponding exterior differential systems. Higherorder constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution (canonical distribution, higherorder Chetaev bundle). With help of a constraint structure a constraint force is naturally introduced. Higherorder mechanical systems subject to different kinds of higherorder constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view. Regular and Lagrangian systems appear as important particular cases within the general scheme.

Reconstruction of the joint timedelay Dopplerscale reflectivity density in the wideband regime. A frame theory based approach
View Description Hide DescriptionThe inverse scattering problem concerning the determination of the joint timedelay Dopplerscale reflectivity density characterizing continuous target environments is addressed by recourse to the generalized frame theory. A reconstruction formula, involving the echoes of a frame of outgoing signals and its corresponding reciprocal frame, is developed. A “realistic” situation with respect to the transmission of a finite number of signals is further considered. In such a case, our reconstruction formula is shown to yield the orthogonal projection of the reflectivity density onto a subspace generated by the transmitted signals.

Dynamics of axial channeling in quasicrystals: An averagingtheory approach
View Description Hide DescriptionA mathematically rigorous Hamiltonian theory of nonrelativistic axial channeling of positively charged particles in simply decorated icosahedral quasicrystals (IQCs) is developed in this paper on the basis of firstorder averaging theory. The main result is an error estimate for the approximation of replacing the relevant Hamiltonian by that of the corresponding axialcontinuum model to calculate suitable phasespace orbits. The derivation of this result makes essential use of a rigorous version of a theorem of Besjes on singlephase firstorder averaging theory and of an asymptotic formula for the distribution of quasilattice points along arbitrary quasilattice axes of the considered IQC model. A deep numbertheoretic result of Niederreiter is used to obtain this formula.

Bering’s proposal for boundary contribution to the Poisson bracket
View Description Hide DescriptionIt is shown that the Poisson bracket with boundary terms proposed by Bering can be deduced from the Poisson bracket proposed by the present author if one omits terms free of Euler–Lagrange derivatives (“annihilation principle”). This corresponds to another definition of the formal product of distributions (or, saying it in other words, to another definition of the pairing between 1forms and 1vectors in the formal variational calculus). We extend the formula (initially suggested by Bering for the ultralocal case with constant coefficients only) onto the general nonultralocal brackets with coefficients depending on fields and their spatial derivatives. The lack of invariance under changes of dependent variables (field redefinitions) seems to be a drawback of this proposal.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


The Markovian limit in a nonlinear quantum kinetic theory
View Description Hide DescriptionWithin the framework of a nonlinear quantum kinetic theory for dissipative farfromequilibrium systems, based on a nonequilibrium ensemble formalism, a rigorous derivation of the Markovian limit is given. This is done in the framework of the nonequilibrium statistical operator method, and resorting to Zubarev’s approach.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


On the Miura Map between the dispersionless KP and dispersionless modified KP hierarchies
View Description Hide DescriptionThe Miura map between the dispersionless KP and dispersionless modified KP hierarchies is investigated. It is shown that the Miura map is canonical with respect to their biHamiltonian structures. Moreover, inspired by the works of Takasaki and Takebe, the twistor construction of solution structure for the dispersionless modified KP hierarchy is given.

Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations
View Description Hide DescriptionIt has been proposed by Hornos and Hornos [Phys. Rev. Lett. 71, 4401–4404 (1993)] that the degeneracy of the genetic code, i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In their work, this picture was developed in the framework of simple Lie algebras. Here, we explore the possibility of explaining the degeneracy of the genetic code using basic classical Lie superalgebras, whose representation theory is sufficiently well understood, at least as far as typical representations are concerned. In the present paper, we give the complete list of all typical codon representations (typical 64dimensional irreducible representations), whereas in the second part, we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code.

Lie superalgebras and the multiplet structure of the genetic code. II. Branching schemes
View Description Hide DescriptionContinuing our attempt to explain the degeneracy of the genetic code using basic classical Lie superalgebras, we present the branching schemes for the typical codon representations (typical 64dimensional irreducible representations) of basic classical Lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code, based on the orthosymplectic algebraosp(52) and differing only in details of the symmetry breaking pattern during the last step.
