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Dec 1998

Volume 39, Issue 12, pp. 6247-6756

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Wigner distribution function for finite systems

Natig M. Atakishiyev, Sergey M. Chumakov, and Kurt Bernardo Wolf

J. Math. Phys. 39, 6247 (1998); http://dx.doi.org/10.1063/1.532636 (15 pages) | Cited 35 times

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We 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 N = 2l+1 data points. The Wigner distribution function assigns classical c-numbers to the operators of position, momentum, and wave guide mode. © 1998 American Institute of Physics.
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42.79.Gn Optical waveguides and couplers
02.50.-r Probability theory, stochastic processes, and statistics

Resolution of some mathematical problems arising in the relativistic treatment of the S states of three-electron systems

D. Matthew Feldmann, Paul J. Pelzl, and Frederick W. King

J. Math. Phys. 39, 6262 (1998); http://dx.doi.org/10.1063/1.532637 (14 pages) | Cited 9 times

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Some of the mathematical difficulties that arise in the evaluation of the Breit–Pauli relativistic energy corrections for the S states of three-electron 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 well-known auxiliary functions. © 1998 American Institute of Physics.
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02.30.Cj Measure and integration
31.30.J- Relativistic and quantum electrodynamic (QED) effects in atoms, molecules, and ions
31.30.Gs Hyperfine interactions and isotope effects

Local U(2,2) symmetry in relativistic quantum mechanics

Felix Finster

J. Math. Phys. 39, 6276 (1998); http://dx.doi.org/10.1063/1.532638 (15 pages) | Cited 9 times

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Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2,2) gauge transformations acting on the spinor index of the wave functions. This local U(2,2) symmetry allows a unified description of electrodynamics and general relativity as a classical gauge theory. © 1998 American Institute of Physics.
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04.50.-h Higher-dimensional gravity and other theories of gravity
04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields

Stochastically positive structures on Weyl algebras. The case of quasi-free states

R. Gielerak, L. Jakóbczyk, and R. Olkiewicz

J. Math. Phys. 39, 6291 (1998); http://dx.doi.org/10.1063/1.532639 (38 pages)

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We consider quasi-free 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 quasi-free 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 so-called thermal process. Then we show that the thermal process completely determines modular structure canonically associated with the quasi-free 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. © 1998 American Institute of Physics.
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02.50.Ey Stochastic processes
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
11.10.Cd Axiomatic approach
02.10.-v Logic, set theory, and algebra
05.30.-d Quantum statistical mechanics
03.70.+k Theory of quantized fields

Gibbs states for AF algebras

Valentin Ya. Golodets and Sergey V. Neshveyev

J. Math. Phys. 39, 6329 (1998); http://dx.doi.org/10.1063/1.532640 (16 pages) | Cited 2 times

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We consider a special class of C∗-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 well-known Araki’s results for a one-dimensional 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 K-systems. © 1998 American Institute of Physics.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
75.10.Jm Quantized spin models, including quantum spin frustration
02.10.-v Logic, set theory, and algebra

Matrix elements for a generalized spiked harmonic oscillator

Richard L. Hall, Nasser Saad, and Attila B. von Keviczky

J. Math. Phys. 39, 6345 (1998); http://dx.doi.org/10.1063/1.532641 (8 pages) | Cited 25 times

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Closed form expressions for the singular-potential integrals 〈m∣x−α∣n〉 are obtained with respect to the Gol’dman and Krivchenkov eigenfunctions for the singular potential Bx2+A/x2, B>0, A ≥ 0. The formulas obtained are generalizations of those found earlier by use of the odd solutions of the Schrödinger equation with the harmonic oscillator potential [Aguilera-Navarro et al., J. Math. Phys. 31, 99 (1990)]. © 1998 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

On infravacua and the localization of sectors

Walter Kunhardt

J. Math. Phys. 39, 6353 (1998); http://dx.doi.org/10.1063/1.532642 (11 pages) | Cited 3 times

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A 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. © 1998 American Institute of Physics.
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12.20.Ds Specific calculations

Coexistent observables and effects in a convexity approach

Pekka Lahti, Sylvia Pulmannova, and Kari Ylinen

J. Math. Phys. 39, 6364 (1998); http://dx.doi.org/10.1063/1.532643 (8 pages) | Cited 3 times

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We 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. © 1998 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Db Functional analytical methods

Strict quantization of coadjoint orbits

N. P. Landsman

J. Math. Phys. 39, 6372 (1998); http://dx.doi.org/10.1063/1.532644 (12 pages) | Cited 1 time

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A strict quantization of a compact symplectic manifold S on a subset I⊆math, containing 0 as an accumulation point, is defined as a continuous field of C∗-algebras {mathℏ}ℏ ∊ I, with math0 = C0(S), and a set of continuous cross sections {Q(f)}f ∊ C∞(S) for which Q0(f) = f. Here Qℏ(f∗) = Qℏ(f)∗ for all ℏ ∊ I, whereas for ℏ→0 one requires that i[Qℏ(f),Qℏ(g)]/ℏ→Qℏ({f,g}) 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 Oλ of a compact connected Lie group G, associated to a highest weight λ. Here I = 0∪1/math, so that ℏ = 1/k, k ∊ math, and math1/k is defined as the C∗-algebra of all matrices on the finite-dimensional Hilbert space Vkλ carrying the irreducible representation Ukλ(G) with highest weight kλ. The quantization maps Q1/k(f) are constructed from coherent states in Vkλ, and have the special feature of being positive maps. © 1998 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.Sv Lie algebras of Lie groups

Deformations of massless Gupta–Bleuler triplets in 3+2 De Sitter space

Marc Lesimple

J. Math. Phys. 39, 6384 (1998); http://dx.doi.org/10.1063/1.532645 (8 pages) | Cited 2 times

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We 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. © 1998 American Institute of Physics.
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11.15.-q Gauge field theories
11.30.Ly Other internal and higher symmetries

Solitary excitations of a two-dimensional electron gas

A. Nerses and E. E. Kunhardt

J. Math. Phys. 39, 6392 (1998); http://dx.doi.org/10.1063/1.532655 (11 pages) | Cited 2 times

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The nonlinear collective excitations of a two-dimensional 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 wave-vector plasmon–polariton regime. © 1998 American Institute of Physics.
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73.21.-b Electron states and collective excitations in multilayers, quantum wells, mesoscopic, and nanoscale systems
71.45.Gm Exchange, correlation, dielectric and magnetic response functions, plasmons
71.36.+c Polaritons (including photon-phonon and photon-magnon interactions)
73.20.Mf Collective excitations (including excitons, polarons, plasmons and other charge-density excitations)

Transition probabilities between quasifree states

H. Scutaru

J. Math. Phys. 39, 6403 (1998); http://dx.doi.org/10.1063/1.532656 (13 pages) | Cited 7 times

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We obtain a general formula for the transition probabilities between any state of the C∗ algebra of the canonical commutation relations (CCR-algebra) 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 one-mode CCR-algebra 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 half-plane 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. © 1998 American Institute of Physics.
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42.50.Dv Quantum state engineering and measurements
02.10.-v Logic, set theory, and algebra

Symmetry adapted states for Hubbard clusters

O. Tjernberg

J. Math. Phys. 39, 6416 (1998); http://dx.doi.org/10.1063/1.532657 (8 pages) | Cited 1 time

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The 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. © 1998 American Institute of Physics.
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71.10.Fd Lattice fermion models (Hubbard model, etc.)
75.10.Lp Band and itinerant models

The q-harmonic oscillator in a lattice model

Hans van Leeuwen and Hans Maassen

J. Math. Phys. 39, 6424 (1998); http://dx.doi.org/10.1063/1.532633 (17 pages) | Cited 1 time

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We give an explicit proof of the pair partitions formula for the moments of the q-harmonic oscillator, and of the claim made by Parisi that the q-deformed lattice Laplacian on the d-dimensional lattice tends to the q-harmonic oscillator in distribution for d→∞. © 1998 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.50.Cw Probability theory

q-phase-coherent states and their squeezing properties

Yaping Yang, Zhixin Lin, Shuangyuan Xie, Weiguo Feng, and Xiang Wu

J. Math. Phys. 39, 6441 (1998); http://dx.doi.org/10.1063/1.532658 (13 pages)

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Some kinds of q-phase-coherent states of a q-harmonic osscillator in a finite-dimensional Hilbert space are constructed. Some properties of these states are discussed. Second-order squeezing properties of these states with respect to the phase quadrature operators are studied. The number-phase squeezing and number-phase uncertainty relations are also studied in detail for a two-state system. Some new number-phase minimum uncertainty states are found. © 1998 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
42.50.Dv Quantum state engineering and measurements

On separable Schrödinger–Maxwell equations

Renat Zhdanov and Maxim Lutfullin

J. Math. Phys. 39, 6454 (1998); http://dx.doi.org/10.1063/1.532659 (5 pages) | Cited 3 times

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We obtain the most general time-dependent potential V(t,x1,x2) 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. © 1998 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.50.De Classical electromagnetism, Maxwell equations
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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

Goong Chen, Sze-Bi Hsu, and Jianxin Zhou

J. Math. Phys. 39, 6459 (1998); http://dx.doi.org/10.1063/1.532670 (31 pages) | Cited 24 times

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A wave equation on a one-dimensional 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 I, energy is injected into the system through a pair of transmission conditions in the feedback form of anti-damping. 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 non-invertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middle-of-the-span 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. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.60.Cb Numerical simulation; solution of equations
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Information gain within nonextensive thermostatistics

Lisa Borland, Angel R. Plastino, and Constantino Tsallis

J. Math. Phys. 39, 6490 (1998); http://dx.doi.org/10.1063/1.532660 (12 pages) | Cited 46 times

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We discuss the information theoretical 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 H-theorem is proved within the generalized context. © 1998 American Institute of Physics.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics
05.70.Ce Thermodynamic functions and equations of state

The quantum canonical ensemble

Dorje C. Brody and Lane P. Hughston

J. Math. Phys. 39, 6502 (1998); http://dx.doi.org/10.1063/1.532661 (7 pages) | Cited 10 times

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The phase space Γ of quantum mechanics can be viewed as the complex projective space CPn endowed with a Kählerian structure given by the Fubini-Study 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 Γ-ensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field. © 1998 American Institute of Physics.
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05.30.Ch Quantum ensemble theory

Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness

Veni Choi, Yong Moon Park, and Hyun Jae Yoo

J. Math. Phys. 39, 6509 (1998); http://dx.doi.org/10.1063/1.532662 (28 pages) | Cited 2 times

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We study the Dirichlet forms and the associated Dirichlet operators for Gibbs measures on infinite particle configuration space. The Dirichlet forms are defined to be “gradient-type” 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 pre-Dirichlet operator is symmetric and that the closure of the pre-Dirichlet form satisfies the Markovian property. When the interaction is three times differentiable and decreasing sub-exponentially, we show that the Dirichlet operator is essentially self-adjoint on a domain consisting of bounded smooth local functions. © 1998 American Institute of Physics.
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05.70.Ce Thermodynamic functions and equations of state
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.30.Hq Ordinary differential equations
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Chaos in the Gyldén problem

Florin Diacu and Dan Şelaru

J. Math. Phys. 39, 6537 (1998); http://dx.doi.org/10.1063/1.532663 (10 pages) | Cited 7 times

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We consider the Gyldén problem—a perturbation of the Kepler problem via an explicit function of time. For certain general classes of planar periodic perturbations, after proving a Poincaré–Melnikov-type criterion, we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi-infinite sequences with infinitely many symbols. We achieve the main result by computing the Melnikov integral explicitly. © 1998 American Institute of Physics.
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95.10.Ce Celestial mechanics (including n-body problems)
05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

A generalized Hirota equation in 2+1 dimensions

Attilio Maccari

J. Math. Phys. 39, 6547 (1998); http://dx.doi.org/10.1063/1.532664 (5 pages) | Cited 21 times

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Using an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, a new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained starting from the Kadomtsev–Petviashvili equation. We apply the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation and demonstrate the integrability property of the new equation, because we obtain the corresponding Lax pair. The new equation reduces to the Hirota equation in the 1+1-dimensional limit. © 1998 American Institute of Physics.
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02.30.Jr Partial differential equations
02.30.Cj Measure and integration
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Bifurcations of new eigenvalues for the Benjamin–Ono equation

Dmitry E. Pelinovsky and Catherine Sulem

J. Math. Phys. 39, 6552 (1998); http://dx.doi.org/10.1063/1.532665 (21 pages) | Cited 6 times

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A criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the Benjamin–Ono (BO) equation. This bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the BO equation. The asymptotic approximation of an exponentially small new eigenvalue is derived. The method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions. Explicit expressions are obtained for the soliton potentials. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.-f Function theory, analysis
41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
11.80.-m Relativistic scattering theory
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Timelike infinity and asymptotic symmetry

Uchida Gen and Tetsuya Shiromizu

J. Math. Phys. 39, 6573 (1998); http://dx.doi.org/10.1063/1.532666 (20 pages) | Cited 2 times

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By extending Ashtekar and Romano’s definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike three-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat space–times at timelike infinity. Asymptotic flatness is characterized by the falloff rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what space–times are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than ∼ t−2, the space–time is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to ∂t falls off at the rate ∼ t−2. It also admits an asymptotic symmetry group similar to the Poincaré group. If the energy-momentum tensor falls off at a rate faster than ∼ t−3, the four-momentum of a space–time may be defined. On the other hand, angular momentum is defined only for space–times in which the energy-momentum tensor falls off at a rate faster than ∼ t−4. © 1998 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
95.30.Sf Relativity and gravitation
97.60.-s Late stages of stellar evolution (including black holes)

Simplicial quantum gravity in the elongated phase

Gabriele Gionti

J. Math. Phys. 39, 6593 (1998); http://dx.doi.org/10.1063/1.532667 (10 pages)

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We discuss the elongated phase of four-dimensional simplicial quantum gravity by exploiting recent analytical results. In particular using Walkup’s theorem we prove that the dominating configurations in the elongated phase are treelike structures called “stacked spheres.” Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyze the critical behavior of theory in the weak coupling regime. © 1998 American Institute of Physics.
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04.60.-m Quantum gravity
98.80.Jk Mathematical and relativistic aspects of cosmology
98.80.Qc Quantum cosmology
95.30.Sf Relativity and gravitation
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