The 2(2S+1)- Formalism and Its Connection with Other Descriptions

In the framework of the Joos-Weinberg 2(2S+1)- theory for massless particles, the dynamical invariants have been derived from the Lagrangian density which is considered to be a 4- vector. A la Majorana interpretation of the 6- component "spinors", the field operators of S=1 particles, as the left- and right-circularly polarized radiation, leads us to the conserved quantities which are analogous to those obtained by Lipkin and Sudbery. The scalar Lagrangian of the Joos-Weinberg theory is shown to be equivalent to the Lagrangian of a free massless field, introduced by Hayashi. As a consequence of a new "gauge" invariance this skew-symmetric field describes physical particles with the longitudinal components only. The interaction of the spinor field with the Weinberg's 2(2S+1)- component massless field is considered. New interpretation of the Weinberg field function is proposed. KEYWORDS: quantum electrodynamics, Lorentz group representation, high-spin particles, bivector, electromagnetic field potential. PACS: 03.50.De, 11.10.Ef, 11.10.Qr, 11.17+y, 11.30.CpComment: 13pp., merged hep-th/9305141 and hep-th/9306108 with revisions. Accepted in "Int. J. Geom. Meth. Phys.

A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.Comment: First presented in Proc. of The 9th Int. Conf. on Clifford Algebras and their Applications, (2011

Axiomatic geometric formulation of electromagnetism with only one axiom: the field equation for the bivector field F with an explanation of the Trouton-Noble experiment

In this paper we present an axiomatic, geometric, formulation of electromagnetism with only one axiom: the field equation for the Faraday bivector field F. This formulation with F field is a self-contained, complete and consistent formulation that dispenses with either electric and magnetic fields or the electromagnetic potentials. All physical quantities are defined without reference frames, the absolute quantities, i.e., they are geometric four dimensional (4D) quantities or, when some basis is introduced, every quantity is represented as a 4D coordinate-based geometric quantity comprising both components and a basis. The new observer independent expressions for the stress-energy vector T(n)(1-vector), the energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the field equation for F. The local conservation laws are also directly derived from that field equation. The 1-vector Lagrangian with the F field as a 4D absolute quantity is presented; the interaction term is written in terms of F and not, as usual, in terms of A. It is shown that this geometric formulation is in a full agreement with the Trouton-Noble experiment.Comment: 32 pages, LaTex, this changed version will be published in Found. Phys. Let

Spin Gauge Theory of Gravity in Clifford Space: A Realization of Kaluza-Klein Theory in 4-Dimensional Spacetime

A theory in which 4-dimensional spacetime is generalized to a larger space, namely a 16-dimensional Clifford space (C-space) is investigated. Curved Clifford space can provide a realization of Kaluza-Klein theory. A covariant Dirac equation in curved C-space is explored. The generalized Dirac field is assumed to be a polyvector-valued object (a Clifford number) which can be written as a superposition of four independent spinors, each spanning a different left ideal of Clifford algebra. The general transformations of a polyvector can act from the left and/or from the right, and form a large gauge group which may contain the group U(1)xSU(2)xSU(3) of the standard model. The generalized spin connection in C-space has the properties of Yang-Mills gauge fields. It contains the ordinary spin connection related to gravity (with torsion), and extra parts describing additional interactions, including those described by the antisymmetric Kalb-Ramond fields.Comment: 57 pages; References added, section 2 rewritten and expande

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related waveComment: Latex fil

Tourist maps - definition, types and contents

Tourist maps are one of the most common groups of cartographic documents. Their variety in terms of content, subject matter and publication titles is a result of growing popularity of diverse forms of tourism activity. The aim of the authors of this article is to demonstrate issues related to tourist maps, including their variety in relation to contemporary forms of tourism. As tourist maps are constantly developing, the authors decided to propose a classification of tourist maps which is adequate from the point of view of the types of maps we currently distinguish. Taking into consideration the aim and type of tourism, the maps were divided into the following sub-groups: maps for sightseeing tourism, qualified tourism, and other tourism, as well as tourist city maps, and maps prepared for promotion and advertising of tourism. The first there categories were further divided into more detailed sub-categories and each of them was described briefly in terms of its content. The classification of maps based on their scales and form of content presentation was also included. The authors attempted also to define the concept of “tourist map” itself. The authors defined it as a geographic representation of an area presented on a plane, in accordance with specific mathematical rules, which should include topographic contents, information about tourist attractions of a given area, its tourist and complementary infrastructure, presented with the help of conventional signs, in a matter appropriate for the scale of the map and its intended use. Contributing to the discussion on the place of tourist map in the general classification of maps, the article distinguishes not only general-geographic maps and thematic maps, but also orientation and navigation maps. This terms covered tourist maps, road maps, and navigation maps: sailing, sea, aerial and city maps. They consist a group of maps in which the functions they play determine their informational content and their form of cartographic presentation. However, unlike on thematic maps, where the general geographic content is merely a background for presentation of the theme-related phenomena, the geographic content is essential in case of tourist maps. It is precisely the general geographic content which is primarily responsible for communicating information which is meant to be used for orientation and navigation purposes