Stochastic Ornstein–Uhlenbeck Capacitors

We introduce and study a class of random capacitor systems which are both charged and discharged stochastically. A capacitor is fed by a random inflow with stationary and independent increments. Discharging occurs according to a Markovian rate which is linear in the capacitors level. The resulting capacitor dynamics are Markovian, stochastically cyclic, and regenerative. We coin these systems Lévy-charged Ornstein–Uhlenbeck capacitors. Various random quantities associated with these systems are analyzed, including: the time-to- discharge; the duration of the charging cycle; the trajectory and the peak height of the capacitor level during a charging cycle; and, the capacitors stationary equilibrium level. Furthermore, we show that there are sharp distinctions between these capacitor systems and corresponding standard Lévy-driven Ornstein–Uhlenbeck systems.     

Crossing relations derived from (Extended) Relativity

Recently, Special Relativity has been straightforwardly extended to Superluminal inertial frames and faster-than-light objects. The Extended Relativity theory not only allowed building up a self-consistent classical theory of tachyons, but reveals itself useful also for the understanding of standard (subluminal) physics, i.e. of usual particles. In this paper, it is shown that Extended Relativity allows: (i) deriving the usual Crossing Relations of elementary particle (high-energy) physics; and (ii) deriving the CPT-covariance theorem as a particular case of G-covariance (i.e., covariance under the new group of Generalised Lorentz transformations, both subluminal and Superluminal).In this framework, the Analyticity postulate is unnecessary: it is better substituted by the G-covariance requirement.Moreover, new crossing-type relations are predicted on the basis of mere Extended Relativity. They may well serve as a test for relativistic covariance of force fields like strong interactions and, particularly, weak interactions, and possible new interaction fields (whicha priori are not relativistically covariant).     

Eigenresonance states in the Kronig-Penney model

We investigate a sandwich of three layer systems with Dirac -functions in the Kronig-Penney model. The inner system ofN=5 atomic layers is enclosed by the two outer systems with different potential strength. The numberM of the atomic layers in the outer system is varied betweenM=9 and infinity, whereas the numberN of the inner layers is held fixed. We obtain the transmission coefficient for the finite system in the region of scattering energies (E>0). An alternating set of transmission gaps, transmission bands and bands of eigenresonance states is obtained. The normalizable eigenresonances occur (forM going to infinity), if a transmission band of the inner system overlaps a transmission gap of the outer systems. The reason for obtaining solutions of standing waves in the band of eigenresonances is the rapid change of the wave phase of a traveling wave, which occurs in a transmission band of the inner system.     

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem

We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V ₁ (x), V ₂ (y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (windows), of lengths equal to the degrees of the potentials V ₁ and V ₂ , satisfy systems of ODEs with polynomial coefficients as well as PDEs (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec     

Rate equations and transient phenomena in semiconductor lasers

This contribution attempts to review certain resonance effects which occur in the dynamic behaviour of semiconductor lasers. To study these effects theoretically, a rate equation approach is used for single-mode operation in the region of lasing threshold.The two basic rate equations are given and their transient solutions discussed. The existence of two time constants in these equations, viz. the electron lifetime _e and the photon lifetime _e; gives rise to a characteristic resonance frequency in the GHz region. This resonance manifests itself in transient spiking effects, in quantum noise phenomena, and in high-frequency modulation experiments. In a modified form the resonance frequency may also be studied in lasers with external cavities and in double-diode configurations (or, equivalently, conventional devices with non-uniform excitation along the cavity length).In the latter two examples mentioned above, the resonance is excited by optical feedback of the laser radiation into the active medium via a lossy or insufficiently inverted region. In the spiking oscillations commonly observed at the commencement of laser operation, the initial population overshoot is the cause of the resonance. For the case of quantum noise it is the requirement that the photon and electron populations have integer values which supplies the driving force-a true quantum effect. High-frequency modulation experiments directly reveal the same resonance frequency where a strong maximum in modulation intensity occurs.A part of this work was reported at the Conference on Semiconductor Injection Lasers and their Applications, Cardiff, March 1972.     

The stress tensor in QED

The gauge-invariant point-splitting version of the energy-momentum tensor in QED is discussed with respect to Poincaré identities. In one-loop approximation an anomaly in the conservation law has been found for the triangle graph.This is part of the author's thesis, Tubingen 1974.     

Colored noise driven systems with inertia

We present a novel approximation scheme, termed unified colored noise approximation (UCNA), for colored Gaussian noise driven nonlinear systems with inertia. This approximation allows one to evaluate static (stationary distributions, moments) as well as dynamical quantities (correlation functions) for small-to-moderate-to-large values of the correlation time. The approximation replaces a three-dimensional Markovian process by a reduced, two-dimensional Markovian dynamics with new drift and diffusion coefficients. For a harmonic potential the stationary moments are reproduced exactly. Most importantly, we present a criterion involving the noise strengthD, the friction strength and the noise color , which describes the region of validity of UCNA in the parameter space given by (D, , ). At small -values we contrast the UCNA with the well-known small approximation. In order to have a comparison onanalytical grounds, we test the static and dynamical predictions of UCNA versus the well-known analytical results obtained from a three-dimensional Ornstein-Uhlenbeck process.     

Semidensities on Odd Symplectic Supermanifolds

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new -operator action on semidensities as the proper framework for the Batalin-Vilkovisky (BV) formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batalin-Vilkovisky geometry. Another application is to (1.1)-codimensional surfaces in E. We construct a kind of pull-back of semidensities to such surfaces. This operation and the -operator are used for obtaining integral invariants for (1.1)-codimensional surfaces.     

On the phase transition of the Ising gauge model

We study the Ising gauge model in dimensionsD2. In addition to the plaquette coupling of this model we introduce a nearest neighbour ferromagnetic Ising coupling. By that means the local gauge symmetry is removed while the global symmetry of the Ising model is still present. The phase diagram of the model is found by using general arguments as well as Monte-Carlo methods. The gauge phase of the model is characterized by the presence of a topological excitation (defect) which has no influence on any local physical quantity.     

Design,fabrication and characterization of high-speed asymmetric Fabry-Perot modulators for optical interconnect applications

Free-space smart-pixel optical interconnect architectures promise to relieve the interconnect bottleneck in high-speed parallel computers and switching systems. One of the most promising output devices that has been advanced for use in these systems is the asymmetric Fabry-Perot modulator, or AFPM, which offers high on-off contrast, low insertion loss, and low operating voltage swing, among other advantages. In this paper we summarize our work on optimizing the AFPM for high-speed operation, including analysis of the material structure design considerations, fabrication of small (16×20 m) devices, and high-speed electrical and optical characterization of the finished modulators. We conclude that at relatively high incident optical intensities the modulators' speed appears to be limited by transit effects to about 18 GHz, but that at lower optical intensities their frequency response outstrips that of our 20 GHz measurement apparatus — that is, these AFPMs are still capable of large signal modulation (20 dB contrast, 1.5 dB insertion loss) at low AC voltage swings (±3 V) for operating frequencies up to 20 GHz. We presume that further investigation will prove them to be RC-limited in this low-intensity regime to speeds of about 35 GHz.     

Noncommutative geometry and a class of completely integrable models

We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (-models or principal chiral models) is then extended to a class of noncommutative harmonic maps into matrix algebras.     

Two-dimensional traveling wave patterns in nonequilibrium systems

Traveling wave patterns formed at the oscillatory onset of convection in binary mixtures with free slip and permeable horizontal boundary conditions are theoretically derived by means of an order parameter equation for the case of large aspect ratio systems with circular and rectangular geometry. We obtain in addition to Zipper states the so called confined states, which so far has been observed only experimentally. Our present study of traveling wave patterns in circular systems, which are the first theoretical investigations in this respect, exhibit a characteristic change of behaviour. Close to onset the waves travel in radial direction towards the sidewalls. For higher Rayleigh-numbers the waves are confined to the circular boundary traveling in azimuthal direction. The occurrence of this transition should be confirmed experimentally.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

Representations of (1,0) and (2,0) Superconformal Algebras in Six Dimensions: Massless and Short Superfields

We construct unitary representations of (1,0) and (2,0) superconformal algebras in six dimensions by using superfields defined on harmonic superspaces with coset manifolds USp(2n)/[U(1)]ⁿ, n=1, 2. In the spirit of the AdS₇/CFT₆ correspondence, massless conformal fields correspond to supersingletons in AdS₇. By tensoring them we produce all short representations corresponding to 1/2 and 1/4 BPS anti-de Sitter bulk states of which massless bulk representations are particular cases.     

Probability interpretation of particle(s) along classical trajectories

The theory of divergent series is used to show that the probability density of a wavemechanical state becomes classical trajectories (for a single particle or many interacting particles) in the limit 0. The probability interpretation is claimed to be valid even if a classical particle idea is inserted into the theory. The idea particle is claimed as a classical entity valid for 0. Finally, the double slit experiment is illustrated as an example and interpreted by the above theory. The interpretation is that the interaction between the incident and interference beams and the double slit is approximately described by the wave theory (interference mechanism).     

An $\mathsf{\epsilon}$-expansion for small-world networks

I construct a well-defined expansion in for diffusion processes on small-world networks. The technique permits one to calculate the average over disorder of moments of the Greens function, and is used to calculate the average Greens function and fluctuations to first non-leading order in , giving results which agree with numerics. This technique is also applicable to other problems of diffusion in random media.Received: 28 July 2004, Published online: 14 December 2004PACS: 89.75.Hc Networks and genealogical trees 64.60.Ak Renormalization-group studies of phase transitions     

Asymptotic Completeness for Compton Scattering

Scattering in a model of a massive quantum-mechanical particle, an electron, interacting with massless, relativistic bosons, photons, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of photons by the electron; but electron-positron pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed electron is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the electron and the photons.Dedicated to Freeman Dyson on the occasion of his 80th birthdayWork partially supported by U.S. National Science Foundation grant DMS 01-00160.Acknowledgement. We thank V. Bach for his hospitality at the University of Mainz, where part of this work was done, and we are indebted to Gian Michele Graf for pointing out a serious gap in an earlier version of this paper. We also thank one of the referees for pointing out many typos and some small errors.     

The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet?

Several situations, in which an empty wave causes an observable effect, are reviewed. They include an experiment showing surrealistic trajectories proposed by Englert et al. and protective measurement of the density of the quantum state. Conditions for observable effects due to empty waves are derived. The possibility (in spite of the existence of these examples) of minimalistic interpretation of Bohmian quantum mechanics in which only Bohmian positions supervene on our experience is discussed.     

Quantum fog and the degradation of information by the gravitational field

In this paper we discuss how information transferred optically through a gravitational field is degraded as the quanta interact with the madium (vacuum state). We quantify information by means of Shannon's entropy, and consider information carriers that are quanta of some field. Next, we obtain the quantum noise (quantum fog) produced by the gravitational field and derive the appropriate channel capacity formula, which quantifies the maximum amount of information that can be transmitted per pulse, in the face of this noise. We show that the channel capacity formula vanishes if the source of information is a space-time singularity because a very intense noise is produced in the vicinity of the singularity. In other words, space-time singularities are hidden behind a very intense quantum fog and cannot be optically observed. A second consequence is that information is degraded as anisotropies (lumpiness) develop in the universe.     

Some remarks on braided group reconstruction and braided doubles

The cross coproduct braided group AutC)B is obtained by Tannaka-Krein reconstruction from C ^B C for a braided group B in braided category C. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain AutC) Aut(C) Aut(C) Aut(C) and higher braided group spin chains. The example B(R) B(R) ... B(R) is described explicitly by R-matrix relations. We also obtain Aut(C) Aut(C)* as a dual quasitriangular codouble braided group by reconstruction from the dual category C° C. General braided double crossproducts B C are also considered.