Special Theory of Relativity without special assumptions and tachyonic motion

The most general form of transformations of space-time coordinates in Special Theory of Relativity based solely on physical assumptions is described. Only the linearity of space-time transformations and the constancy of the speed of light are used as assumptions. The application to tachyonic motion is indicated.Описано найзагальнішу форму перетворень координат простір - час у спеціальній теорії відносності, що базується виключно на фізичних припущеннях. Як припущення використано тільки лінійність перетворень простір - час і постійність швидкості світла. Вказано на застосування до тахіонного руху

Galilei invariant theories. I. Constructions of indecomposable finite-dimensional representations of the homogeneous Galilei group: directly and via contractions

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations are also obtained via contractions of the corresponding representations of the Lorentz group. Finally the obtained representations are used to derive a general Pauli anomalous interaction term and Darwin and spin-orbit couplings of a Galilean particle interacting with an external electric field.Comment: 23 pages, 2 table

On the electrodynamics of moving bodies at low velocities

We discuss the seminal article in which Le Bellac and Levy-Leblond have identified two Galilean limits of electromagnetism, and its modern implications. We use their results to point out some confusion in the literature and in the teaching of special relativity and electromagnetism. For instance, it is not widely recognized that there exist two well defined non-relativistic limits, so that researchers and teachers are likely to utilize an incoherent mixture of both. Recent works have shed a new light on the choice of gauge conditions in classical electromagnetism. We retrieve Le Bellac-Levy-Leblond's results by examining orders of magnitudes, and then with a Lorentz-like manifestly covariant approach to Galilean covariance based on a 5-dimensional Minkowski manifold. We emphasize the Riemann-Lorenz approach based on the vector and scalar potentials as opposed to the Heaviside-Hertz formulation in terms of electromagnetic fields. We discuss various applications and experiments, such as in magnetohydrodynamics and electrohydrodynamics, quantum mechanics, superconductivity, continuous media, etc. Much of the current technology where waves are not taken into account, is actually based on Galilean electromagnetism

A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\bar\Psi},{\Psi}] = G_0[{\lambda\bar\Psi}, {\lambda\Psi}] + const., 0<\lambda\in R (\bar\Psi_k, \Psi_k, k=1,...,n, are the generators of the Grassmann algebra G_2n), between the finite-dimensional analogues G_0 and G of the (``classical'') action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If \lambda \not= 1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If \lambda=1, the considered Grassmann integral equation has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.Comment: 58 pages LaTeX (v2: mainly, minor updates and corrections to the reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54], [61], [64], [139] added

Imaging of the lenses of the human eye by ultrabiomicroscopy (UBM), ultrasonography (USG) and by anterior segment optical coherence tomography (AS OCT): - presentation of clinical cases

The nervous system's ability to receive light stimuli and its' processing in the brain in order to produce a visual impression is the definition of the sense of sight. The anatomical form of the sense organ of vision is comprised of the eyeball, the eye's protective apparatus, the eye's movement apparatus and the retinal nerve connections made to structures in the brain. The shape of the eye's lens, which gives the eye its refraction ability, depends on the voltage present in Zinn's ligaments that regulate the ciliary muscle. Sharpness of vision is produced by changing the shape of the lens, a reflexive adjustment. A domed lens causes stronger light inflexion and allows a sharp visual appearance of close objects. A flattening of the lens results in less light refraction and the seeing of more distant objects. The lens consists of a capsule, a cortex and a nucleus and it has two convex surfaces: the front and the rear. If we imagine the lens as a plum fruit, the capsule is its skin, the cortex is its flesh and the nucleus is its stone. Proper functioning of the lens is essential for accurate vision. Exact assessments of the eye's lens and the ability to monitor the status of associated diseases are extremely important. The lens may be tested using a slit lamp, but in any situation where an eye disease prevents this assessment of the lens this examination will obviously be inadequate. Thanks to today's imaging techniques, we can now assess the anatomical arrangement and condition of the lens, even in the eyes of those patients for whom the use of an imaging lens slit lamp is not possible [1, 2]