Information theory applied to space-time physics

Description

The aim of this book is to throw light on various facets of geometry through development of four geometrical themes. The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations.The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincaré disc model of the hyperbolic plane is developed 1. Historical Review. 1.1. Roots in the Greek World. 1.2. From Euclidean to Non-Euclidean Geometry. 1.3. Metric and Differential Geometry. 1.4. Physical Space-Time -- 2. Information Theory Applied to Measurements. 2.1. The Concept of Information. 2.2. Finite Information and Finite Resolution. 2.3. Finite Information Flow. 2.4. Finite Resolution [Delta]x and [Delta]t -- 3. Coordinate Systems. 3.1. Coordinate Systems Based on Rings. 3.2. The Belief in Three-Dimensional Space. 3.3. Right and Left Handed Structures. 3.4. Distance in Discrete Coordinate Systems. 3.5. Coordinate Systems Defined by the Metric Tensor. 3.6. Coordinate Systems with one Variable -- 4. Time and Motion. 4.1. Time and Time Differences. 4.2. Shifts and Propagation. 4.3. Velocity. 4.4. Three Time Dimensions and One Space Dimension -- 5. Propagation in Unusual Coordinate Systems. 5.1. Dyadic Metric. 5.2. Dyadic Coordinate Systems. 5.3. Standing Waves and Topology. 5.4. Observed Shifts and Their Eigenfunctions. 5.5. Nondyadic Coordinate Systems. 5.6. Motion Based on Integer Number and Dyadic Topology. 5.7. Dyadic Clocks -- 6. Distinction of Sinusoidal Functions. 6.1. Differential and Partial Equations. 6.2. Concepts of the Calculus of Finite Differences. 6.3. Concepts of the Dyadic Calculus. 6.4. Difference Calculus on General Rings. 6.5. Spectral Decomposition of Light. 6.6. Laser for Nonsinusoidal Waves -- 7. Discrete Topologies and Difference Equations. 7.1. Finite Differences and Differentials. 7.2. Dyadic Difference Quotient. 7.3. Observable Effects of Discrete Topologies -- 8. Schrodinger and Klein-Gordon Difference Equations. 8.1. Time Dependence of the Solutions of Partial Difference Equations. 8.2. Schrodinger Equation. 8.3. Physical Meaning of a Difference Equation. 8.4. Klein-Gordon Equation -- 9. Schrodinger Difference Equation with Coulomb Field. 9.1. Separation of Variables for a Centrally Symmetric Field. 9.2. Discrete Eigenvalues in a Coulomb Field. 9.3. Time Variable Solutions -- 10. Klein-Gordon Difference Equation with Coulomb Field. 10.1. Separation of Variables and Initial Value Problem. 10.2. Discrete Eigenvalues of Bosons in a Coulomb Field. 10.3. Asymptotic Solution. 10.4. Convergent Solution. 10.5. Convergence at the Origin. 10.6. Free Particles in a Coulomb Field. 10.7. Independence Relation -- 11. Dirac Difference Equation with Coulomb Field. 11.1. Iterated Dirac Equation. 11.2. Linearized Dirac Equation -- 12. Mathematical Supplements. 12.1. Transmission Capacity. 12.2. Distributive Law for Dyadic Multiplication. 12.3. Method for Dyadic Division. 12.4. Right and Left Difference Quotient. 12.5. Independence Relation in Three-Dimensional Cartesian Coordinates. 12.6. Polynomials as Solutions of Difference Equations of Second Order. 12.7. Difference Equation of the Discrete Spherical Harmonics. 12.8. Convergence of the Solution of the Klein-Gordon Equation with Coulomb Field. 12.9. Orthogonality of Eigenfunctions. 12.10. Generalization of Green's Formula. 12.11. Eigenfunctions of the Dyadic Difference Operator