Volume 39, Issue 5, May 1998
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Gamow vectors for degenerate scattering resonances
View Description Hide DescriptionIn this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation.

Representationtheoretic aspects of twodimensional quantum systems in singular vector potentials: Canonical commutation relations, quantum algebras, and reduction to lattice quantum systems
View Description Hide DescriptionSome representationtheoretic aspects of a twodimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of are investigated. For each vector v in a set the projection of the physical momentum operator to the direction of v is defined by as an operator acting in where with (resp., ) being the generalized partial differential operator in the variable (resp., ) and is a parameter denoting the charge of the particle. It is proven that is essentially selfadjoint and an explicit formula is derived for the strongly continuous oneparameter unitary group generated by the selfadjoint operator (the closure of ), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in are also considered. Conjugately to and a selfadjoint multiplication operator is introduced, which is a linear combination of the position operators and such that, if A is flat on then gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation are analyzed. In particular, a necessary and sufficient condition for to be unitarily equivalent (or inequivalent) to the Schrödinger representation of CCR is established. The case where is inequivalent to the Schrödinger representation corresponds to the Aharonov–Bohm effect. Quantum algebraic structures [quantum plane and the quantum group] associated with the pair are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice [the case ], where and are linearly independent, it is shown that the magnetic translations with A replaced by a modified vector potential are reduced by the Hilbert space identified with a closed subspace of This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones.

Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping
View Description Hide DescriptionCanonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik–Zamolodchikov equations. One can solve these equations by the offshell Bethe ansatz method provided the Knizhnik–Zamolodchikov equations are related with the highest weight representations of the corresponding Lie algebras: These solutions can be written in terms of multivariable generalizations of special functions of the hypergeometric type. In this work, we consider a realization of the above scheme for the Maxwell–Bloch system with pumping: quantum states for this system are found in terms of the multivariable confluent hypergeometric function.

Geometry of the Batalin–Fradkin–Vilkovisky theorem
View Description Hide DescriptionWe describe gauge fixing at the level of virtual paths in the path integral as a nonsymplectic BRSTtype of flow on the path phase space. As a consequence, a gaugefixed, nonlocal symplectic structure arises. Restoring of locality is discussed. A pertinent antiLiebracket and an infinite dimensional group of gauge fermions are introduced. Generalizations to Sp(2)symmetric BLT theories are made.

Imaginary parts of Stark–Wannier resonances
View Description Hide DescriptionWe consider a onedimensional Stark–Wannier Hamiltonian, where is a smooth periodic, finitegap potential, and is small enough. We compute rigorously the imaginary parts of the spectral resonances. For this purpose we develop some related elements of the adiabatic approach to the equations of the form

An exactly soluble Schrödinger equation with smooth positiondependent mass
View Description Hide DescriptionThe onedimensional generalized Schrödinger equation for a system with smooth potential and mass step is resolved exactly. The wave function depends on the Heun’s function, which is a solution of a secondorder Fuchsian equation with four singularities. The behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step. Two limiting cases are also studied: when the width of the mass step is vanishing, and when the smooth potential and mass step tend to an abrupt potential and mass step.

Intermediate level statistics with oneparameter random matrix ensembles
View Description Hide DescriptionWe discuss a formulation of the level statistics for quantum systems that lie in an intermediate regime between the integrability subject to Poisson statistics and the full chaos subject to Gaussian statistics of the typical random matrix theory (RMT). It is based on the idea initiated by Yukawa and also by Nakamura and Lakshmanan, namely, to transcribe the eigenvalue statistics into statistical mechanics of the completely integrable Calogero system with internal degrees of rotation. By analyzing the previous works of Gaudin and Forrester from this viewpoint, we answer the question raised by Mehta and Dyson in early RMT concerning the expectation of compressible level gas, which is realized now in such an intermediate regime. It indicates the general absence of a fractional power dependence of the twolevel correlation function under the ordinary thermodynamiclimit procedure as the manifestation that a transition between localized and delocalized states does not occur. The predictability of the transition by modifying the theory is remarked. Two applications of the present scheme are shown: the information loss per level and the correlation hole of survival probability for intermediate circular unitary ensembles.

Path integral approach to noncommutative space–times
View Description Hide DescriptionWe propose a path integral formulation of noncommutative generalizations of space–time manifold in even dimensions, characterized by a length scale The commutative case is obtaind in the limit

From Turbiner’s quasiexactly soluble potentials in dimensions to analytic solutions of the combined Coulomb plus oscillator system
View Description Hide DescriptionTurbiner’s onedimensional, quasiexactly soluble potentials can be interpreted as central potentials in dimensions. Using a particular transformation, one shows how these potentials convert to seemingly unrelated isolated solutions of other Schrödinger equations, including solutions of the combined Coulomb plus harmonic oscillator system.

Quantum averaging. III. Timedependent systems
View Description Hide DescriptionWe present new perturbation algorithms for timedependent quantum systems. These new methods make use of a timedependent averaging method and are based on analogies with the Poincaré–von Zeipel (PvZ) and Kolmogorov–Arnold–Moser (KAM) expansions. Application to a simple example shows that the PvZ as well as the KAM expansion are superior to the standard quantum mechanical timedependent perturbation method (Dyson expansion).

A uniqueness theorem for distributions and its application to nonlocal quantum field theory
View Description Hide DescriptionIt is shown that if a distribution has support in a convex cone not containing a straight line and its Fourier transform is carried by a closed cone different from the whole space, i.e., then is zero. This result leads to a new and most general proof of the PCT theorem which covers nonlocal quantum fields with arbitrary highenergy behavior.

Standard forms of local nets in quantum field theory
View Description Hide DescriptionNets of von Neumann algebras, with local algebras associated with all compact, convex, causally complete subsets of spacetime, are constructed, assuming a particular kind of net considered by Bisognano and Wichmann. The properties of such nets are discussed, within the general framework of algebraic quantum field theory, with an emphasis on the relation between the geometry of spacetime and the structure of the nets. An “intersection property” for the local algebras is stated and proved. The structure of subnets of such nets is discussed. A number of particular possible features of the nets are shown to be hereditary in the sense that the subnets have the same features. Possible further extensions of the domain of definition of these nets are discussed briefly.

Hyperfinitedimensional representations of canonical commutation relation
View Description Hide DescriptionThis paper presents some methods of representing canonical commutation relations in terms of hyperfinitedimensional matrices, which are constructed by nonstandard analysis. The first method uses representations of a nonstandard extension of the finite Heisenberg group, called hyperfinite Heisenberg group. The second is based on hyperfinitedimensional representations of Then, the cases of infinite degree of freedom are argued in terms of the algebra of hyperfinite paraFermi oscillators, which is mathematically equivalent to a hyperfinitedimensional representation of

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Solutions of the modified chiral model in (2+1) dimensions
View Description Hide DescriptionIn this paper we deal with classical solutions of the modified chiral model on Such solutions are shown to correspond to products of various factors which we call timedependent unitons. Then the problem of solving the system of secondorder partial differential equations for the chiral field is reduced to solving a sequence of systems of firstorder partial differential equations for the unitons.

Alternative potentials for the electromagnetic field
View Description Hide DescriptionThe sourcefree electromagnetic field can be expressed in terms of two complex potentials α,β, which are related to the Debye potentials. The potentials {α,β} corresponding to various fields are discussed. A conserved radiation density ρ can be constructed in terms of these potentials, this density is positive (negative) for positive (negative) helicity radiation.

Hamiltonian timedependent mechanics
View Description Hide DescriptionThe usual formulation of timedependent mechanics implies a given splitting of an event space This splitting, however, is broken by any timedependent transformation, including transformations between inertial frames. The goal is the framecovariant formulation of timedependent mechanics on a bundle whose fibration is not fixed. Its phase space is the vertical cotangent bundle provided with the canonical 3form and the corresponding canonical Poissonstructure. An event space of relativistic mechanics is a manifold whose fibration is not fixed.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


New structures in the theory of the laser model. II. Microscopic dynamics and a nonequilibrium entropy principle
View Description Hide DescriptionIn a recent article, Alli and Sewell [J. Math. Phys. 36, 5598 (1995)] formulated a new version of the Dicke–Hepp–Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time maximizes its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.

Diffusion on braided spaces
View Description Hide DescriptionThe notion of qBrownian motion introduced by Majid is extended to braided spaces corresponding to a generic Rmatrix, and combined with the theory of quantum probability. This leads to a definition of diffusions on these spaces. The corresponding heat equations (differencedifferential equations) are solved in terms of Appell polynomials (i.e., shifted moment systems). Some examples of interest for applications are given.

Brownian motion of quantum harmonic oscillators: Existence of a subdynamics
View Description Hide DescriptionThe effects of systemenvironment correlations on the dynamics of an open quantum system are investigated for the standard model of a set of quantum harmonic oscillators interacting with a heat bath of oscillators. By definition, a subdynamics is described by transformations of the open system observables. It is shown that such a construction can reproduce the observable properties of the exact dynamics only when the states of system and environment are uncorrelated, while for classical systems there is always a subdynamics. A quantum subdynamics cannot have the properties we associate with thermal fluctuations; the KMS relation at a finite temperature for the open system implies that the system must be closed. The conditions for having a subdynamics as a good approximation to the exact closed dynamics are investigated, and so are the similar but stronger conditions for a Markovian dynamics. It is also shown that a subdynamics defines the response of the open system to some types of time dependent external forces.

Study of hydrodynamical limits in a multicollision scale Boltzmann equation for semiconductors
View Description Hide DescriptionWe derive in this paper a generalized hydrodynamical model for semiconductors from a multicollision scale Boltzmann equation. We apply a Chapman–Enskog method for two timescale collisional operators. These two time scales are linked respectively with the elastic part of the electron/lattice scattering term and the electron/electron collision term. Our model has a global entropy function and a symmetric formulation. It is a generalization of the classical hydrodynamical model since we add two terms: a “thermal friction force” and a “friction heat flux.” Moreover, with some changes of parameters we obtain both the energytransport model and the Euler–Poisson hydrodynamical model.
