Volume 39, Issue 12, December 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Wigner distribution function for finite systems
View Description Hide DescriptionWe construct a Wigner distribution function for finite data sets. It is based on a finite optical system; a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry. The dynamical group for this model is SU(2) and the wave functions are sets of data points. The Wigner distribution function assigns classical cnumbers to the operators of position, momentum, and wave guide mode.

Resolution of some mathematical problems arising in the relativistic treatment of the S states of threeelectron systems
View Description Hide DescriptionSome of the mathematical difficulties that arise in the evaluation of the Breit–Pauli relativistic energy corrections for the S states of threeelectron systems are resolved. Evaluation of the expectation value of the Breit–Pauli Hamiltonian using explicitly correlated wave functions leads to sets of integrals that diverge individually. By appropriately combining these integrals, and using some judicious series expansions, all the integration problems are resolved in terms of wellknown auxiliary functions.

Local symmetry in relativistic quantum mechanics
View Description Hide DescriptionLocal gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local gauge transformations acting on the spinor index of the wave functions. This local symmetry allows a unified description of electrodynamics and general relativity as a classical gauge theory.

Stochastically positive structures on Weyl algebras. The case of quasifree states
View Description Hide DescriptionWe consider quasifree stochastically positive ground and thermal states on Weyl algebras in the imaginary time formulation. In particular, we obtain a new derivation of a general form of thermal quasifree state and give conditions when such a state is stochastically positive, i.e., when it defines a periodic stochastic process with respect to imaginary time, a socalled thermal process. Then we show that the thermal process completely determines modular structure canonically associated with the quasifree thermal state on Weyl algebra. We discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space–times and models of quantum statistical mechanics.

Gibbs states for AF algebras
View Description Hide DescriptionWe consider a special class of systems containing asymptotically Abelian binary shifts and shifts of Temperley–Lieb algebras. We study Gibbs states for these systems corresponding to potentials with finite range interaction, and obtain the same results as the wellknown Araki’s results for a onedimensional quantum lattice. In particular, it is proved that a Gibbs state in the infinite volume is a translation invariant KMS state having the exponential uniform clustering property. Entropic properties of the Gibbs states are also discussed. This allows us, in particular, to construct new examples of quantum Ksystems.

Matrix elements for a generalized spiked harmonic oscillator
View Description Hide DescriptionClosed form expressions for the singularpotential integrals are obtained with respect to the Gol’dman and Krivchenkov eigenfunctions for the singular potential The formulas obtained are generalizations of those found earlier by use of the odd solutions of the Schrödinger equation with the harmonic oscillator potential [AguileraNavarro et al., J. Math. Phys. 31, 99 (1990)].

On infravacua and the localization of sectors
View Description Hide DescriptionA certain class of superselection sectors of the free massless scalar field in three space dimensions is considered. It is shown that these sectors, which cannot be localized with respect to the vacuum, acquire a much better localization, namely in spacelike cones, when viewed in front of suitable “infravacuum” backgrounds. These background states coincide, essentially, with a class of states introduced by Kraus, Polley, and Reents as models for clouds of infrared radiation.

Coexistent observables and effects in a convexity approach
View Description Hide DescriptionWe characterize functionally coexistent observables in terms of biobservables and joint observables. Coexistent sets of effects are characterized in terms of projective systems of simple observables.

Strict quantization of coadjoint orbits
View Description Hide DescriptionA strict quantization of a compact symplectic manifoldS on a subset , containing 0 as an accumulation point, is defined as a continuous field of algebras with , and a set of continuous cross sections for which . Here for all , whereas for one requires that in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit of a compact connected Lie groupG, associated to a highest weight . Here , so that , , and is defined as the algebra of all matrices on the finitedimensional Hilbert space carrying the irreducible representation with highest weight . The quantization maps are constructed from coherent states in and have the special feature of being positive maps.

Deformations of massless Gupta–Bleuler triplets in De Sitter space
View Description Hide DescriptionWe study deformations of massless indecomposable representations (Gupta–Bleuler triplets) for the De Sitter group SO(3,2), showing that there is no deformation of such representations toward a kind of massive Gupta–Bleuler triplets which would have corresponded to an analog of a Higgs–Kibble mechanism in anti De Sitter space.

Solitary excitations of a twodimensional electron gas
View Description Hide DescriptionThe nonlinear collective excitations of a twodimensional electron gas(2DEG) formed at the interface of a heterostructure are presented. A matrix formulation of the coupled particle dynamics–electromagnetic field equations permits the extraction of the equation of evolution for these excitations. The stationary solutions of the equation are presented. A new class of solitary excitations is shown to form part of the nonlinear mode spectrum of excitations of the 2DEG in the low wavevector plasmon–polariton regime.

Transition probabilities between quasifree states
View Description Hide DescriptionWe obtain a general formula for the transition probabilities between any state of the algebra of the canonical commutation relations (CCRalgebra) and a squeezed quasifree state (Theorem III.1). Applications of this formula are made for the case of multimode thermal squeezed states of quantum optics using a general canonical decomposition of the correlation matrix valid for any quasifree state. In the particular case of a onemode CCRalgebra we show that the transition probability between two quasifree squeezed states is a decreasing function of the geodesic distance between the points of the upper halfplane representing these states. In the special case of the purification map it is shown that the transition probability between the state of the enlarged system and the product state of real and fictitious subsystems can be a measure for the entanglement.

Symmetry adapted states for Hubbard clusters
View Description Hide DescriptionThe application of spin, pseudo spin and space group symmetry to find symmetry adapted states for the square planar Hubbard model is discussed. An approach based on pseudo spin configurations and the application of Young tableaux to the permutation group is presented. The method is illustrated for the case of a 4^{*}4 cluster and the complete classification of the states for this cluster is given. It is shown that the linear dimension of the largest matrix block is reduced by three orders of magnitude by application of the above symmetries.

The qharmonic oscillator in a lattice model
View Description Hide DescriptionWe give an explicit proof of the pair partitions formula for the moments of the qharmonic oscillator, and of the claim made by Parisi that the qdeformed lattice Laplacian on the ddimensional lattice tends to the qharmonic oscillator in distribution for

qphasecoherent states and their squeezing properties
View Description Hide DescriptionSome kinds of qphasecoherent states of a qharmonic osscillator in a finitedimensional Hilbert space are constructed. Some properties of these states are discussed. Secondorder squeezing properties of these states with respect to the phase quadrature operators are studied. The numberphase squeezing and numberphase uncertainty relations are also studied in detail for a twostate system. Some new numberphase minimum uncertainty states are found.

On separable Schrödinger–Maxwell equations
View Description Hide DescriptionWe obtain the most general timedependent potential enabling separating of variables in the (1+2)dimensional Schrödinger equation. With the use of this result the four classes of separable Schrödinger–Maxwell equations are constructed.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span
View Description Hide DescriptionA wave equation on a onedimensional interval I has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is fixed. At exactly the midpoint of the interval energy is injected into the system through a pair of transmission conditions in the feedback form of antidamping. We wish to study chaotic wave propagation in the system. A cause of chaos by snapback repellers has been identified. These snapback repellers are repelling fixed points possessing homoclinic orbits of the noninvertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middleofthespan points. Existing literature [F. R. Marotto, J. Math. Anal. Appl. 63, 199–223 (1978)] on snapback repellers contains an error. We clarify the error and give a refined theorem that snapback repellers imply chaos. Numerical simulations of chaotic vibration are also illustrated.
 Top

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Information gain within nonextensive thermostatistics
View Description Hide DescriptionWe discuss the informationtheoretical foundations of the Kullback information gain, recently generalized within a nonextensive thermostatistical formalism. General properties are studied and, in particular, a consistent test for measuring the degree of correlation between random variables is proposed. In addition, minimum entropy distributions are discussed and the Htheorem is proved within the generalized context.

The quantum canonical ensemble
View Description Hide DescriptionThe phase space Γ of quantum mechanics can be viewed as the complex projective space endowed with a Kählerian structure given by the FubiniStudy metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin onehalf particle in a heat bath with an applied magnetic field.

Dirichlet forms and Dirichlet operators for infinite particle systems: Essential selfadjointness
View Description Hide DescriptionWe study the Dirichlet forms and the associated Dirichlet operators for Gibbs measures on infinite particle configuration space. The Dirichlet forms are defined to be “gradienttype” forms by introducing a measurable field of rigged Hilbert spaces on the configuration space. Under mild conditions on the interaction including singular potentials, we show that the preDirichlet operator is symmetric and that the closure of the preDirichlet form satisfies the Markovian property. When the interaction is three times differentiable and decreasing subexponentially, we show that the Dirichlet operator is essentially selfadjoint on a domain consisting of bounded smooth local functions.
