Contracted Representation of Yang's Space-Time Algebra and Buniy-Hsu-Zee's Discrete Space-Time

Motivated by the recent proposition by Buniy, Hsu and Zee with respect to discrete space-time and finite spatial degrees of freedom of our physical world with a short- and a long-distance scales, $l_P$ and $L,$ we reconsider the Lorentz-covariant Yang's quantized space-time algebra (YSTA), which is intrinsically equipped with such two kinds of scale parameters, $\lambda$ and $R$. In accordance with their proposition, we find the so-called contracted representation of YSTA with finite spatial degrees of freedom associated with the ratio $R/\lambda$, which gives a possibility of the divergence-free noncommutative field theory on YSTA. The canonical commutation relations familiar in the ordinary quantum mechanics appear as the cooperative Inonu-Wigner's contraction limit of YSTA, $\lambda \to 0$ and $R \to \infty.

Noncommutative space-time models

The FRT quantum Euclidean spaces $O_q^N$ are formulated in terms of Cartesian generators. The quantum analogs of N-dimensional Cayley-Klein spaces are obtained by contractions and analytical continuations. Noncommutative constant curvature spaces are introduced as a spheres in the quantum Cayley-Klein spaces. For N=5 part of them are interpreted as the noncommutative analogs of (1+3) space-time models. As a result the quantum (anti) de Sitter, Newton, Galilei kinematics with the fundamental length and the fundamental time are suggested.Comment: 8 pages; talk given at XIV International Colloquium of Integrable Systems, Prague, June 16-18, 200

Quantum Theory and Galois Fields

We discuss the motivation and main results of a quantum theory over a Galois field (GFQT). The goal of the paper is to describe main ideas of GFQT in a simplest possible way and to give clear and simple arguments that GFQT is a more natural quantum theory than the standard one. The paper has been prepared as a presentation to the ICSSUR' 2005 conference (Besancon, France, May 2-6, 2005).Comment: Latex, 24 pages, 1 figur

Wigner's little group, gauge transformations and dimensional descent

We propose a technique called dimensional descent to show that Wigner's little group for massless particles, which acts as a generator of gauge transformation for usual Maxwell theory, has an identical role even for topologically massive gauge theories. The examples of $B\wedge F$ theory and Maxwell-Chern-Simons theory are analyzed in details.Comment: LaTex, revised version shortened to 9 pages; To appear in Jour.Phys.

The language of Einstein spoken by optical instruments

Einstein had to learn the mathematics of Lorentz transformations in order to complete his covariant formulation of Maxwell's equations. The mathematics of Lorentz transformations, called the Lorentz group, continues playing its important role in optical sciences. It is the basic mathematical language for coherent and squeezed states. It is noted that the six-parameter Lorentz group can be represented by two-by-two matrices. Since the beam transfer matrices in ray optics is largely based on two-by-two matrices or $ABCD$ matrices, the Lorentz group is bound to be the basic language for ray optics, including polarization optics, interferometers, lens optics, multilayer optics, and the Poincar\'e sphere. Because the group of Lorentz transformations and ray optics are based on the same two-by-two matrix formalism, ray optics can perform mathematical operations which correspond to transformations in special relativity. It is shown, in particular, that one-lens optics provides a mathematical basis for unifying the internal space-time symmetries of massive and massless particles in the Lorentz-covariant world.Comment: LaTex 8 pages, presented at the 10th International Conference on Quantum Optics (Minsk, Belarus, May-June 2004), to be published in the proceeding

Coherent States and N Dimensional Coordinate Noncommutativity

Considering coordinates as operators whose measured values are expectations between generalized coherent states based on the group SO(N,1) leads to coordinate noncommutativity together with full $N$ dimensional rotation invariance. Through the introduction of a gauge potential this theory can additionally be made invariant under $N$ dimensional translations. Fluctuations in coordinate measurements are determined by two scales. For small distances these fluctuations are fixed at the noncommutativity parameter while for larger distances they are proportional to the distance itself divided by a {\em very} large number. Limits on this number will lbe available from LIGO measurements.Comment: 16 pqges. LaTeX with JHEP.cl

Higgsless Electroweak Model and Contraction of Gauge Group

A modified formulation of the Electroweak Model with 3-dimensional spherical geometry in the target space is suggested. The {\it free} Lagrangian in the spherical field space along with the standard gauge field Lagrangian form the full Higgsless Lagrangian of the model, whose second order terms reproduce the same experimentally verified fields with the same masses as the Standard Electroweak Model. The vector bosons masses are automatically generated, so there is no need in special mechanism of spontaneous symmetry breaking. The limiting case of the modified Higgsless Electroweak Model, which corresponds to the contracted gauge group $SU(2;j)\times U(1)$ is discussed. Within framework of the limit model Z-boson, electromagnetic and electron fields are interpreted as an external ones with respect to W-bosons and neutrino fields. The W-bosons and neutrino fields do not effect on these external fields. The masses of all particles remain the same, but the field interactions in contracted model are more simple as compared with the standard Electroweak Model due to nullification of some terms.Comment: Talk at the International Workshop "`Supersymmetries and Quantum Symmetries"' (SQS-09), Dubna, Russia, July 29 -- August 3, 2009, 11

Limiting Case of Modified Electroweak Model for Contracted Gauge Group

The modification of the Electroweak Model with 3-dimensional spherical geometry in the matter fields space is suggested. The Lagrangian of this model is given by the sum of the {\it free} (without any potential term) matter fields Lagrangian and the standard gauge fields Lagrangian. The vector boson masses are generated by transformation of this Lagrangian from Cartesian coordinates to a coordinates on the sphere $S_3$. The limiting case of the bosonic part of the modified model, which corresponds to the contracted gauge group $SU(2;j)\times U(1)$ is discussed. Within framework of the limit model Z-boson and electromagnetic fields can be regarded as an external ones with respect to W-bosons fields in the sence that W-boson fields do not effect on these external fields. The masses of all particles of the Electroweak Model remain the same, but field interactions in contracted model are more simple as compared with the standard Electroweak Model.Comment: 12 pages, talk given at the XIII Int. Conf. on SYMMETRY METHODS IN PHYSICS, Dubna, Russia, July 6-9, 2009; added references for introduction, clarified motivatio

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