Volume 41, Issue 11, November 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


A discretevelocity, stationary Wigner equation
View Description Hide DescriptionThis paper is concerned with the onedimensional stationary linear Wigner equation, a kinetic formulation of quantum mechanics. Specifically, we analyze the wellposedness of the boundary value problem on a slab of the phase space with given inflow data for a discretevelocity model. We find that the problem is uniquely solvable if zero is not a discrete velocity. Otherwise one obtains a differentialalgebraic equation of index 2 and, hence, the inflow data make the system overdetermined.

Quantum stochastic models of twolevel atoms and electromagnetic cross sections
View Description Hide DescriptionQuantum stochastic differential equations have been used to describe the dynamics of an atom interacting with the electromagnetic field via absorption/emission processes. Here, by using the full quantum stochastic Schrödinger equation proposed by Hudson and Parthasarathy, we show that such models can be generalized to include other processes into the interaction. In the case of a twolevel atom we construct a model on the basis of some physical requirements, the main being a balance equation on the fluxes of the ingoing and outgoing photons; in this model the atomfield interaction turns out to be due either to absorption/emission processes either to direct scattering processes, which simulate the interaction due to virtual transitions to the levels which have been eliminated from the description. To see the effects of the new terms, we consider both direct and heterodyne detection of the fluorescence light emitted by an atom stimulated by a monochromatic coherent laser and we deduce from these two detection schemes the expressions of the total, elastic and inelastic electromagnetic cross sections and the spectral distribution of the fluorescence light. The total cross section, as a function of the frequency of the stimulating laser, can present not only a Lorentzian shape, but the full variety of Fano profiles; intensity dependent widths and shifts are obtained. The fluorescencespectrum can present complicated shapes, according to the values of the various parameters; when the direct scattering is not important the usual symmetric triplet structure of the Mollow spectrum appears (for high intensity of the stimulating laser), while a strong contribution of the direct scattering process can distort such a triplet structure or can even make it to disappear.

Nonlocal regularization for nonAbelian gauge theories for arbitrary gauge parameter
View Description Hide DescriptionWe study the nonlocal regularization for the nonAbelian gauge theories for an arbitrary value of the gauge parameter ξ. We show that the procedure for the nonlocalization of field theories established earlier by the original authors, when applied in that form to the Faddeev–Popov effective action in a linear gauge cannot lead to a ξindependent result for the observables. We then show that an alternate procedure which is simpler can be used and that it leads to the Smatrix elements (where they exist) independent of ξ.

Quantum stochastic differential equation for unstable systems
View Description Hide DescriptionA semiclassical nonHamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes nonHamiltonian at random instants of transition of pure states to reduced ones, given by a contraction The counting trajectories are assumed to satisfy the Poisson law. A unitary dilation of the concractive stochastic dynamics is found. In particular, in the limit of frequent detection corresponding to the large number limit we obtain the Itô–Schrödinger stochastic unitary evolution for the pure state of unstable quantum system providing a new stochastic version of the quantum Zeno effect.

“Composite particles” and the eigenstates of Calogero–Sutherland and Ruijsenaars–Schneider
View Description Hide DescriptionWe establish a onetoone correspondence between the “composite particles” with N particles and the Young tableaux with at most N rows. We apply this correspondence to the models of Calogero–Sutherland and Ruijsenaars–Schneider and we obtain a momentum space representation of the “composite particles” in terms of creation operators attached to the Young tableaux. Using the technique of bosonization, we obtain a position space representation of the “composite particles” in terms of products of vertex operators. In the special case where the “composite particles” are bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series of the fractional quantum Hall effect.

Fractional statistic
View Description Hide DescriptionWe improve Haldane’s formula which gives the number of configurations for N particles on d states in a fractional statistic defined by the coupling Although nothing is changed in the thermodynamic limit, the new formula makes sense for finite with the p integer and A geometrical interpretation of fractional statistic is given in terms of “composite particles.”

Superheating field for the Ginzburg–Landau equations in the case of a large bounded interval
View Description Hide DescriptionWe study the asymptotic behavior of the local superheating field for a film of width in the regime κ small, large, where κ is the Ginzburg–Landau parameter. This gives a mathematical justification for the introduction of the semiinfinite model as a good approximation for this regime.

Geometric models of dimensional relativistic rotating oscillators
View Description Hide DescriptionGeometricmodels of quantum relativistic rotating oscillators in arbitrary dimensions are defined on backgrounds with deformed antide Sitter metrics. It is shown that these models are analytically solvable, deriving the formulas of the energy levels and corresponding normalized energy eigenfunctions. An important property is that all these models have the same nonrelativistic limit, namely the usual harmonic oscillator.

A simple algebraic derivation of the covariant anomaly and Schwinger term
View Description Hide DescriptionAn expression for the curvature of the “covariant” determinant line bundle is given in evendimensional space–time. The usefulness is guaranteed by its prediction of the covariant anomaly and Schwinger term. It allows a parallel derivation of the consistent anomaly and Schwinger term, and their covariant counterparts, which clarifies the similarities and differences between them.

Superspace formulation of general massive gauge theories and geometric interpretation of massdependent Becchi–Rouet–Stora–Tyutin symmetries
View Description Hide DescriptionA superspace formulation is proposed for the osp(1,2)covariant Lagrangian quantization of general massive gauge theories. The superalgebra is considered as subalgebra of the superalgebra which may be considered as the algebra of generators of the conformal group in a superspace with two anticommuting coordinates. The massdependent (anti)Becchi–Rouet–Stora–Tyutin symmetries of proper solutions of the quantum master equations in the osp(1,2)covariant formalism are realized in that superspace as invariance under translations combined with massdependent special conformal transformations. The symmetry—in particular the ghost number conservation—and the new ghost number conservation are realized in the superspace as invariance under symplectic rotations and dilatations, respectively. The new ghost number conservation is generally broken by the choice of a gauge. The transformations of the gauge fields and the full set of necessarily required (anti)ghost and auxiliary fields under the superalgebra are determined both for irreducible and firststage reducible theories with closed gauge algebra.

Canonical Noether symmetries and commutativity properties for gauge systems
View Description Hide DescriptionFor a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.

Characterizations of the canonical phase observable
View Description Hide DescriptionIn this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables. We also show that any contractive weighted shift operator defines a unique phase observable, and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit.

On the Kepler problem
View Description Hide DescriptionUsing the idea that the symmetry generators commuting with a Landaulike Hamiltonian containing nonAbelian gauge fields will be matrixvalued differential operators, we reconsider the eigenvalue problem of the fivedimensional (5D) Kepler problem on a instanton background. We quickly reproduce the result of Pletyukhov and Tolkachev [J. Math. Phys. 40, 93–100 (1999)], obtained for the energy spectrum. The eigenstates can be expressed in terms of the monopole harmonics. The relevance of the theory of induced representations for solving similar problems is emphasized.

Exact solution of the quantum Calogero–Gaudin system and of its q deformation
View Description Hide DescriptionA complete set of commuting observables for the Calogero–Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the coalgebra invariance of the model; with the proper technical modifications this procedure can be applied to the qdeformed version of the model, which is then also exactly solved.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Geometry of multisymplectic Hamiltonian firstorder field theories
View Description Hide DescriptionIn the jet bundle description of field theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures (canonical or not) needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyperregular and almostregular cases. Finally, the role of connections in the construction of Hamiltonian field theories is clarified.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Kinetic foundation of extended irreversible thermodynamics
View Description Hide DescriptionIn this paper we investigate the kinetic foundation of extended irreversible thermodynamics via the moment method. First we consider the construct of the 1particle distribution function in terms of its moments by maximizing the entropy density function. We then project from its space onto the local thermodynamic variables in the thermodynamic base space Thus instead of the Boltzmann equation we consider a set of evolution equations of in Second, we formulate the laws of thermodynamics governing the variable in These laws exhibit an intrinsic geometric structure of thermodynamics in the setting of contact geometry. Finally, as an illustration, we discuss the evolution equations for the bulk pressure heat flux Q, and the symmetric traceless tensor corresponding to the viscous and heat conduction irreversible processes. These equations can be formulated as an abstract inhomogeneous hyperbolic evolution equation. By employing the semigroup technique, we discuss the solution of the evolution equation and its asymptotic behavior. We show that thermodynamic stability condition of the system implies asymptotic dynamical stability of the solution and vice versa.

Kadanoff–Baym equations and nonMarkovian Boltzmann equation in generalized Tmatrix approximation
View Description Hide DescriptionA recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized Tmatrix approximation for the selfenergies. It is shown that the Tmatrix obtains additional contributions arising from initial correlations. Using these results and taking the timediagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a farreaching generalization of Boltzmanntype kinetic equations: It selfconsistently includes memory effects (retardation, offshell Tmatrices) as well as manyparticle effects (damping, inmedium Tmatrices) and spinstatistics effects (Pauliblocking).
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Putting an edge to the Poisson bracket
View Description Hide DescriptionWe consider a general formalism for treating a Hamiltonian (canonical) field theory with a spatial boundary. In this formalism essentially all functionals are differentiable from the very beginning and hence no improvement terms are needed. We introduce a new Poisson bracket which differs from the usual “bulk” Poisson bracket with a boundary term and show that the Jacobi identity is satisfied. The result is geometrized on an abstract world volume manifold. The method is suitable for studying systems with a spatial edge like the ones often considered in Chern–Simons theory and General Relativity. Finally, we discuss how the boundary terms may be related to the time ordering when quantizing.

On the decomposition of the modified Kadomtsev–Petviashvili equation and explicit solutions
View Description Hide DescriptionThe (2+1)dimensional modified Kadomtsev–Petviashvili equation is decomposed into two known (1+1)dimensional soliton equations. A Darboux transformation of the two known (1+1)dimensional soliton equations is derived with the help of a gauge transformation of the spectral problem. As an application, explicit solutions of the two (1+1)dimensional soliton equations and two new explicit solutions of the (2+1)dimensional modified Kadomtsev–Petviashvili equation are obtained.

Families of metrics geodesically equivalent to the analogs of the Poisson sphere
View Description Hide DescriptionThe metric of the Poisson sphere can be obtained as a result of reduction from the kinetic energy of the free motion of a rigid body. It is proved that the metric of the Poisson sphere admits nontrivial oneparameter family of metrics that have the same geodesic lines. In the present paper we prove that the Clebsch case of motion of the rigid body and the case of the free motion of a rigid body in Minkowski space lead (after reduction) to Riemannian metrics that also admit nontrivial geodesical equivalence. A new integrability criterion of geodesical equivalence is proved.
