Principal Fibrations from Noncommutative Spheres

We construct noncommutative principal fibrations S_θ⁷→S_θ⁴ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. “The algebra inclusion is an example of a not-trivial quantum principal bundle.”     

Noncommutative theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer index

Fujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)_A anomaly for noncommutative SU(N). We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θ^μν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

Gauge theories in the many-gluon limit

We argue that the idea that the dynamics of a gauge theory simplifies in the limit N → ∞, where N is the number of colors, can be invoked even if the gauge group is an exceptional Lie group, rather than one of the classical groups. We also point out that quantum tunneling phenomena can in some cases survive in the N → ∞ limit, contrary to the usual claim that the N → ∞ limit is “classical.”     

Wave Node Dynamics and Revival Symmetry in Quantum Rotors

Symmetries and dynamics of wave nodes in space and time expose principles of quantum theory and its relativistic underpinning. Among these are key principles behind recently discovered dephasing and rephasing phenomena known as revivals. A reexamination of basic Eberly revivals, Berry “quantum fractal” landscapes, and the “quantum carpets” of Schleich and co-workers reveals a simple Farey arithmetic and C_n-group revival structure in one of the earliest quantum wave models, the Bohr rotor. These principles may be useful for interpreting modern time-dependent rovibrational spectra. The nodal dynamics of the Bohr rotor is seen to have a quasi-fractal structure similar to that of earlier systems involving chaotic circle maps. The fractal structure is an overlay of discrete versions of Bohr's rotor model. Each N-point Bohr rotor acts like a base-N quantum “odometer” which performs rational fraction arithmetic. Such systems may have applications for optical information technology and quantum computing.     

Highly strained InxGa(1−x)As−InyAl(1−y)As (x>0.8,y<0.3) layers for short wavelength QWIP and QCL structures grown by MBE

Highly strained quantum cascade laser (QCL) and quantum well infrared photodetector (QWIPs) structures based on In_xGa_(1−x)As−In_yAl_(1−y)As (x>0.8,y<0.3) layers have been grown by molecular beam epitaxy. Conditions of exact stoichiometric growth were used at a temperature of 420°C to produce structures that are suitable for both emission and detection in the 2–5 μm mid-infrared regime. High structural integrity, as assessed by double crystal X-ray diffraction, room temperature photoluminescence and electrical characteristics were observed. Strong room temperature intersubband absorption in highly tensile strained and strain-compensated In_0.84Ga_0.16As/AlAs/In_0.52Al_0.48As double barrier quantum wells grown on InP substrates is demonstrated. Γ–Γ intersubband transitions have been observed across a wide range of the mid-infrared spectrum (2–7 μm) in three structures of differing In_0.84Ga_0.16As well width (30, 45, and 80 Å). We demonstrate short-wavelength IR, intersubband operation in both detection and emission for application in QC and QWIP structures. By pushing the InGaAs–InAlAs system to its ultimate limit, we have obtained the highest band offsets that are theoretically possible in this system both for the Γ–Γ bands and the Γ–X bands, thereby opening up the way for both high power and high efficiency coupled with short-wavelength operation at room temperature. The versatility of this material system and technique in covering a wide range of the infrared spectrum is thus demonstrated.     

Central elements of the elliptic monodromy matrix algebra at roots of unity

The central elements of the algebra of monodromy matrices associated with the R-matrix are studied. When the crossing parameter w takes a special rational value , where N and n are positive coprime integers, the center is substantially larger than that in the generic case for which the “quantum determinant” provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo.     

Photoexcited carrier dynamics in aligned InAs/GaAs quantum dots grown on strain-relaxed InGaAs layers

Carrier dynamics in aligned InAs/GaAs quantum dots (QDs) grown on cross-hatched patterns induced by metastable In_xGa_1−xAs layers have been studied by time-resolved photoluminescence. The low-temperature carrier lifetimes were found to be of the order of 100–200 ps and determined by carrier trapping and nonradiative recombination. Comparisons with control “nonaligned” InAs QDs show remarkable differences in dependence of peak PL intensities on excitation power, and in PL decay times dependences on both temperature and excitation intensities. Possible origin of traps, which determine the carrier lifetimes, is discussed.     

Dynamical behavior of the phase transition of strained from atomistic simulations

Using an atomistic shell model we study the temperature dependence of the ferroelectric properties of BaTiO₃ under biaxial compressive strain applicable to growth on perovskites substrate. Molecular dynamics simulations show a “r→c→p” sequence of phase transitions when temperature is increased, and the absence of an “ac phase”. The first-order paraelectric-to-ferroelectric phase transition presents in bulk changes to a second-order one as a consequence of the in-plane constraint imposed by the mechanical boundary conditions. From the tetragonal ferroelectric c phase, the transition takes place in a finite range of temperature where the lattice parameter normal to the plane keeps approximately constant until T_c is reached. Analysis of the local polarization behavior reveals an order–disorder dynamics as the dominant mechanism of the transition.     

Fourier transform emission spectroscopy of the TiF radical in the 407 nm region

Ultraviolet emission spectra of the TiF radical in the 407 nm region have been observed at a resolution of 0.04 cm⁻¹ using a Fourier transform spectrometer. A new electronic assignment of ⁴Γ–X⁴Φ has been proposed. Rotational analysis has been obtained for the 0–0 and 1–1 vibrational bands of the ⁴Γ_5/2–X⁴Φ_3/2, ⁴Γ_9/2–X⁴Φ_7/2, and ⁴Γ_11/2–X⁴Φ_9/2 subbands and the 0–0 band of ⁴Γ_7/2–X⁴Φ_5/2. The lower state rotational and centrifugal distortion constants are consistent with the previous results [J. Mol. Spectrosc. 184 (1997) 186; J. Chem. Phys. 119 (2003) 9496], to the conformation that the lower state of the 407 nm band is the ⁴Φ ground electronic state. Rough estimates of the vibrational interval ΔG(1/2) and the spin–orbit coupling constant A in the ⁴Γ state were also obtained.     

Thermodynamics on noncommutative geometry in coherent state formalism

The thermodynamics of ideal gas on the noncommutative geometry in the coherent state formalism is investigated. We first evaluate the statistical interparticle potential and see that there are residual “attraction (repulsion) potential” between boson (fermion) in the high temperature limit. The characters could be traced to the fact that, the particle with mass m in noncommutative thermal geometry with noncommutativity θ and temperature T will correspond to that in the commutative background with temperature T(1+kTmθ)⁻¹. Such a correspondence implies that the ideal gas energy will asymptotically approach to a finite limiting value as that on commutative geometry at T_θ=(kmθ)⁻¹. We also investigate the squeezed coherent states and see that they could have arbitrary mean energy. The thermal properties of those systems are calculated and compared to each other. We find that the heat capacity of the squeezed coherent states of boson and fermion on the noncommutative geometry have different values, contrast to that on the commutative geometry.     

K–K excitations on as “primary” superfields

We show that the K–K spectrum of IIB string on is described by “twisted chiral” superfields, naturally described in “harmonic superspace”, obtained by taking suitable gauge singlets polynomials of the D3-brane boundary superconformal field theory.To each p-order polynomial is associated a massive K–K short representation with states. The quadratic polynomial corresponds to the “supercurrent multiplet” describing the “massless” bulk graviton multiplet.     

Double-dielectric-mirror InGaN/GaN microcavities formed using selective removal of an AlInN layer

We describe the fabrication and optical properties of a 3λ/2 InGaN/GaN-based microcavity using “upper” and “lower” silica/zirconia mirrors. The fabrication of this structure involved selective removal of an AlInN layer following multistep thinning of a free-standing GaN substrate. Photoluminescence spectra show a narrowing of the excitonic emission from InGaN/GaN quantum wells in the microcavity, giving a cavity quality factor Q exceeding 400.     

Some Graded Bialgebras Related to Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra by wU_q^t(G)wU_{q}^{\tau}(\mathcal{G}). It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra U_q(G)U_{q}(\mathcal{G}) of G\mathcal{G}.     

Magnetooptical studies of Cd1−xMnxTe quantum wells with parabolic confining potential

We report on magnetooptical studies of II–VI semiconductor quantum wells with a parabolic shape of the potential grown on the basis of Cd_1−xMn_xTe. Photoluminescence excitation measurements revealed a series of peaks equidistant in energy associated with interband optical transitions between “harmonic oscillator states”. We observed the Zeeman splitting for heavy-hole excitons up to the subband index n=5. From the comparison of the experimental data with numerical calculations for the Zeeman splitting it was possible to determine the correct shape of the potential.     

States on Quantum Structures Versus Integrals

We show conditions when a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL-algebra, a pseudo MV-algebra and a pseudo BL-algebra is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K. The same is true for states on an MV-algebra and a BL-algebra and their noncommutative variants.     

Quantum mechanics with difference operators

A formulation of quantum mechanics with additive and multiplicative (q-) difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a “natural” choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a “new” framework.     

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.     

The Relation Between Two Types of Characteristic Equations for Quantum Matrices

A definition (modification) of the power of quantum matrices using the -matrix has recently been proven useful to obtain generalizations of many well known theorems from linear algebra to the quantum case, among which are the Cayley–Hamilton theorem and the Newton identities. A separate effort has provided another generalization of the Cayley–Hamilton theorem for GL _q (n), which uses usual matrix powers but diagonal matrices as coefficients.We show that the latter generalization can be derived in the aforementioned more general framework and it is the expression of the modified quantum power in terms of the usual ones that accounts for the appearance of diagonal matrices.