Waves on Noncommutative Spacetimes

Waves on ``commutative'' spacetimes like R^d are elements of the commutative algebra C^0(R^d) of functions on R^d. When C^0(R^d) is deformed to a noncommutative algebra {\cal A}_\theta (R^d) with deformation parameter \theta ({\cal A}_0 (R^d) = C^0(R^d)), waves being its elements, are no longer complex-valued functions on R^d. Rules for their interpretation, such as measurement of their intensity, and energy, thus need to be stated. We address this task here. We then apply the rules to interference and diffraction for d \leq 4 and with time-space noncommutativity. Novel phenomena are encountered. Thus when the time of observation T is so brief that T \leq 2 \theta w, where w is the frequency of incident waves, no interference can be observed. For larger times, the interference pattern is deformed and depends on \frac{\theta w}{T}. It approaches the commutative pattern only when \frac{\theta w}{T} goes to 0. As an application, we discuss interference of star light due to cosmic strings.Comment: 19 pages, 5 figures, LaTeX, added references, corrected typo

Interacting Quantum Topologies and the Quantum Hall Effect

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by A_theta(R^2) x A_theta(R^2) the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding centre coordinates. We argue that A_theta(R^2) itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on A_theta(R^2) to electromagnetic fields which are described by the abelian commutative gauge group G_c(U(1)), i.e. gauge fields based on A_0(R^2). This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras A_theta^prime with different theta^prime appear. Two-particle wave functions in this approach are based on A_theta(R^2) x A_theta(R^2). We find that the full symmetry group in IQHE, which is the semi-direct product SO(2) \ltimes G_c(U(1)) acts on this tensor product using the twisted coproduct Delta_theta. Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.Comment: 10 pages, LaTeX, Corrected typos, Published versio

Spontaneous Breaking of Lorentz Symmetry and Vertex Operators for Vortices

We first review the spontaneous Lorentz symmetry breaking in the presence of massless gauge fields and infraparticles. This result was obtained long time ago in the context of rigorious quantum field theory by Frohlich et. al. and reformulated by Balachandran and Vaidya using the notion of superselection sectors and direction-dependent test functions at spatial infinity for the non-local observables. Inspired by these developments and under the assumption that the spectrum of the electric charge is quantized, (in units of a fundamental charge e) we construct a family of vertex operators which create winding number k, electrically charged Abelian vortices from the vacuum (zero winding number sector) and/or shift the winding number by k units. In particular, we find that for rotating vortices the vertex operator at level k shifts the angular momentum of the vortex by k \frac{{\tilde q}}{q}, where \tilde q is the electric charge of the quantum state of the vortex and q is the charge of the vortex scalar field under the U(1) gauge field. We also show that, for charged-particle-vortex composites angular momentum eigenvalues shift by k \frac{{\tilde q}}{q}, {\tilde q} being the electric charge of the charged-particle-vortex composite. This leads to the result that for \frac{{\tilde q}}{q} half-odd integral and for odd k our vertex operators flip the statistics of charged-particle-vortex composites from bosons to fermions and vice versa. For fractional values of \frac{{\tilde q}}{q}, application of vertex operator on charged-particle-vortex composite leads in general to composites with anyonic statistics.Comment: Published version, 15+1 pages, 1 figur

Spontaneous Lorentz Violation: The Case of Infrared QED

It is by now clear that infrared sector of QED has an intriguingly complex structure. Based on earlier pioneering works on this subject, two of us recently proposed a simple modification of QED by constructing a generalization of the $U(1)$ charge group of QED to the "Sky" group incorporating the known spontaneous Lorentz violation due to infrared photons, but still compatible in particular with locality. There it was shown that the "Sky" group is generated by the algebra of angle dependent charges and a study of its superselection sectors has revealed a manifest description of spontaneous breaking of Lorentz symmetry. We further elaborate this approach here and investigate in some detail the properties of charged particles dressed by the infrared photons. We find that Lorentz violation due to soft photons may be manifestly codified in an angle dependent fermion mass modifying therefore the fermion dispersion relations. The fact that the masses of the charged particles are not Lorentz invariant affects their spin content too.Time dilation formulae for decays should also get corrections. We speculate that these effects could be measured possibly in muon decay experiments.Comment: 18+1 pages, revised version, expanded discussion in section 5

Quantum Hall Effect on the Grassmannians Gr2(CN)\mathbf{Gr}_2(\mathbb{C}^N)

Quantum Hall Effects (QHEs) on the complex Grassmann manifolds $\mathbf{Gr}_2(\mathbb{C}^N)$ are formulated. We set up the Landau problem in $\mathbf{Gr}_2(\mathbb{C}^N)$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner ${\cal D}$-functions for charged particles on $\mathbf{Gr}_2(\mathbb{C}^N)$ under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. In particular, for the simplest case of $\mathbf{Gr}_2(\mathbb{C}^4)$ we explicitly write down the $U(1)$ background gauge field as well as the single and many-particle eigenstates by introducing the Pl\"{u}cker coordinates and show by calculating the two-point correlation function that the Lowest Landau Level (LLL) at filling factor $\nu =1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for QHE on ${\mathbb C}P^N$ and generalize them to all $\mathbf{Gr}_2(\mathbb{C}^N)$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on $\mathbf{Gr}_2(\mathbb{C}^4)$ and the Hall effect on the odd sphere $S^5$, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on ${\mathbb C}P^3$ and ${\mathbb C}P^7$ and those on the spheres $S^4$ and $S^8$, respectively.Comment: 34 pages, revtex 4-1, Minor Correction

Morphological variation of carotid artery bifurcation level in digital angiography

Knowing of the level of carotid artery bifurcation (CB) is important for vascular surgery in the neck, radical neck dissections, carotid sinus baroreceptor stimulation, catheterisations, and aneurysms. The aim of this study was to determine the CB level in relation with the cervical vertebral levels, compare them on the right and the left sides, and investigate the relation of CB level with the length of neck. In this study, 100 conventional carotid angiographies were performed. The CB level was determined in relation with 10 different levels which were the levels of the cervical vertebrae and intervertebral disks, and the relation of CB level with the length of neck was investigated. The right and left CB levels of the patients were also determined, and compared. The highest level of CB was at the level of C2 vertebra, and the lowest level of CB was at the level of C6–C7 intervertebral disk in both male and female. When all patients were taken into consideration, CB level was most frequently seen at the level of C4–C5 (29%) on the right side, and at the level of C4 (26%) on the left side. The CB levels were not symmetrical in 10 female and 23 male. Knowing of the anatomical variations of CB level is important in surgical procedures. The anatomical differences must be taken into consideration since the neighbouring structures of CB change in case of variations. We believe that the results of this study will shed light to planning of all interventional methods concerning common carotid artery and its branches as well as surgery in the neck, and will help to minimise the complications