120 research outputs found
Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I
In two series of papers we construct quasi regular polyhedra and their duals
which are similar to the Catalan solids. The group elements as well as the
vertices of the polyhedra are represented in terms of quaternions. In the
present paper we discuss the quasi regular polygons (isogonal and isotoxal
polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal
hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain
aperiodic tilings of the plane with the isogonal polygons along with the
regular polygons. We point out that one type of aperiodic tiling of the plane
with regular and isogonal hexagons may represent a state of graphene where one
carbon atom is bound to three neighboring carbons with two single bonds and one
double bond. We also show how the plane can be tiled with two tiles; one of
them is the isotoxal polygon, dual of the isogonal polygon. A general method is
employed for the constructions of the quasi regular prisms and their duals in
3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure
Wigner's Spins, Feynman's Partons, and Their Common Ground
The connection between spin and symmetry was established by Wigner in his
1939 paper on the Poincar\'e group. For a massive particle at rest, the little
group is O(3) from which the concept of spin emerges. The little group for a
massless particle is isomorphic to the two-dimensional Euclidean group with one
rotational and two translational degrees of freedom. The rotational degree
corresponds to the helicity, and the translational degrees to the gauge degree
of freedom. The question then is whether these two different symmetries can be
united. Another hard-pressing problem is Feynman's parton picture which is
valid only for hadrons moving with speed close to that of light. While the
hadron at rest is believed to be a bound state of quarks, the question arises
whether the parton picture is a Lorentz-boosted bound state of quarks. We study
these problems within Einstein's framework in which the energy-momentum
relations for slow particles and fast particles are two different
manifestations one covariant entity.Comment: LaTex 12 pages, 3 figs, based on the lectures delivered at the
Advanced Study Institute on Symmetries and Spin (Prague, Czech Republic, July
2001
Fourier-Space Crystallography as Group Cohomology
We reformulate Fourier-space crystallography in the language of cohomology of
groups. Once the problem is understood as a classification of linear functions
on the lattice, restricted by a particular group relation, and identified by
gauge transformation, the cohomological description becomes natural. We review
Fourier-space crystallography and group cohomology, quote the fact that
cohomology is dual to homology, and exhibit several results, previously
established for special cases or by intricate calculation, that fall
immediately out of the formalism. In particular, we prove that {\it two phase
functions are gauge equivalent if and only if they agree on all their
gauge-invariant integral linear combinations} and show how to find all these
linear combinations systematically.Comment: plain tex, 14 pages (replaced 5/8/01 to include archive preprint
number for reference 22
Relativistic nature of a magnetoelectric modulus of Cr_2O_3-crystals: a new 4-dimensional pseudoscalar and its measurement
Earlier, the magnetoelectric effect of chromium sesquioxide Cr_2O_3 has been
determined experimentally as a function of temperature. One measures the
electric field-induced magnetization on Cr_2O_3 crystals or the magnetic
field-induced polarization. From the magnetoelectric moduli of Cr_2O_3 we
extract a 4-dimensional relativistic invariant pseudoscalar
. It is temperature dependent and of the order of
10^{-4}/Z_0, with Z_0 as vacuum impedance. We show that the new pseudoscalar is
odd under parity transformation and odd under time inversion. Moreover,
is for Cr_2O_3 what Tellegen's gyrator is for two port
theory, the axion field for axion electrodynamics, and the PEMC (perfect
electromagnetic conductor) for electrical engineering.Comment: Revtex, 36 pages, 9 figures (submitted in low resolution, better
quality figures are available from the authors
Theory of Superconducting of doped fullerenes
We develop the nonadiabatic polaron theory of superconductivity of
taking into account the polaron band narrowing and realistic
electron-phonon and Coulomb interactions. We argue that the crossover from the
BCS weak-coupling superconductivity to the strong-coupling polaronic and
bipolaronic superconductivity occurs at the BCS coupling constant independent of the adiabatic ratio, and there is nothing ``beyond'' Migdal's
theorem except small polarons for any realistic electron-phonon interaction. By
the use of the polaronic-type function and the ``exact'' diagonalization in the
truncated Hilbert space of vibrons (``phonons'') we calculate the ground state
energy and the electron spectral density of the molecule. This
allows us to describe the photoemission spectrum of in a wide
energy region and determine the electron-phonon interaction. The strongest
coupling is found with the high-frequency pinch mode and with the
Frenkel exciton. We clarify the crucial role of high-frequency bosonic
excitations in doped fullerenes which reduce the bare bandwidth and the Coulomb
repulsion allowing the intermediate and low-frequency phonons to couple two
small polarons in a Cooper pair. The Eliashberg-type equations are solved for
low-frequency phonons. The value of the superconducting , its pressure
dependence and the isotope effect are found to be in a remarkable agreement
with the available experimental data.Comment: 20 pages, Latex, 4 figures available upon reques
The Hubbard model within the equations of motion approach
The Hubbard model has a special role in Condensed Matter Theory as it is
considered as the simplest Hamiltonian model one can write in order to describe
anomalous physical properties of some class of real materials. Unfortunately,
this model is not exactly solved except for some limits and therefore one
should resort to analytical methods, like the Equations of Motion Approach, or
to numerical techniques in order to attain a description of its relevant
features in the whole range of physical parameters (interaction, filling and
temperature). In this manuscript, the Composite Operator Method, which exploits
the above mentioned analytical technique, is presented and systematically
applied in order to get information about the behavior of all relevant
properties of the model (local, thermodynamic, single- and two- particle ones)
in comparison with many other analytical techniques, the above cited known
limits and numerical simulations. Within this approach, the Hubbard model is
shown to be also capable to describe some anomalous behaviors of the cuprate
superconductors.Comment: 232 pages, more than 300 figures, more than 500 reference
Classical Simulation of Relativistic Quantum Mechanics in Periodic Optical Structures
Spatial and/or temporal propagation of light waves in periodic optical
structures offers a rather unique possibility to realize in a purely classical
setting the optical analogues of a wide variety of quantum phenomena rooted in
relativistic wave equations. In this work a brief overview of a few optical
analogues of relativistic quantum phenomena, based on either spatial light
transport in engineered photonic lattices or on temporal pulse propagation in
Bragg grating structures, is presented. Examples include spatial and temporal
photonic analogues of the Zitterbewegung of a relativistic electron, Klein
tunneling, vacuum decay and pair-production, the Dirac oscillator, the
relativistic Kronig-Penney model, and optical realizations of non-Hermitian
extensions of relativistic wave equations.Comment: review article (invited), 14 pages, 7 figures, 105 reference
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