# the geometry of special relativity a concise course springerbriefs in physics

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## The Geometry Of Special Relativity A Concise Course

**Author :**Norbert Dragon

**ISBN :**9783642283291

**Genre :**Science

**File Size :**43. 9 MB

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In this concise primer it is shown that, with simple diagrams, the phenomena of time dilatation, length contraction and Lorentz transformations can be deduced from the fact that in a vacuum one cannot distinguish physically straight and uniform motion from rest, and that the speed of light does not depend on the speed of either the source or the observer. The text proceeds to derive the important results of relativistic physics and to resolve its apparent paradoxes. A short introduction into the covariant formulation of electrodynamics is also given. This publication addresses, in particular, students of physics and mathematics in their final undergraduate year.

## Quantum Field Theory On Curved Spacetimes

**Author :**Christian Bär

**ISBN :**9783642027802

**Genre :**Science

**File Size :**67. 74 MB

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After some decades of work a satisfactory theory of quantum gravity is still not available; moreover, there are indications that the original field theoretical approach may be better suited than originally expected. There, to first approximation, one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes. Ingredients of this approach are the formulation of quantum physics in terms of C*-algebras, the geometry of Lorentzian manifolds, in particular their causal structure, and linear hyperbolic differential equations where the well-posedness of the Cauchy problem plays a distinguished role, as well as more recently the insights from suitable concepts such as microlocal analysis. This primer is an outgrowth of a compact course given by the editors and contributing authors to an audience of advanced graduate students and young researchers in the field, and assumes working knowledge of differential geometry and functional analysis on the part of the reader.

## Geometry Of Minkowski Space Time

**Author :**Francesco Catoni

**ISBN :**3642179770

**Genre :**Science

**File Size :**32. 30 MB

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This book provides an original introduction to the geometry of Minkowski space-time. A hundred years after the space-time formulation of special relativity by Hermann Minkowski, it is shown that the kinematical consequences of special relativity are merely a manifestation of space-time geometry. The book is written with the intention of providing students (and teachers) of the first years of University courses with a tool which is easy to be applied and allows the solution of any problem of relativistic kinematics at the same time. The book treats in a rigorous way, but using a non-sophisticated mathematics, the Kinematics of Special Relativity. As an example, the famous "Twin Paradox" is completely solved for all kinds of motions. The novelty of the presentation in this book consists in the extensive use of hyperbolic numbers, the simplest extension of complex numbers, for a complete formalization of the kinematics in the Minkowski space-time. Moreover, from this formalization the understanding of gravity comes as a manifestation of curvature of space-time, suggesting new research fields.

## Quantum Physics Fuzzy Sets And Logic

**Author :**Jarosław Pykacz

**ISBN :**9783319193847

**Genre :**Science

**File Size :**25. 78 MB

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This Brief presents steps towards elaborating a new interpretation of quantum mechanics based on a specific version of Łukasiewicz infinite-valued logic. It begins with a short survey of main interpretations of quantum mechanics already proposed, as well as various models of many-valued logics and previous attempts to apply them for the description of quantum phenomena. The prospective many-valued interpretation of quantum mechanics is soundly based on a theorem concerning the isomorphic representation of Birkhoff-von Neumann quantum logic in the form of a special Łukasiewicz infinite-valued logic endowed with partially defined conjunctions and disjunctions.

## Entanglement Measures And Their Properties In Quantum Field Theory

**Author :**Stefan Hollands

**ISBN :**3319949012

**Genre :**Science

**File Size :**61. 38 MB

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This book gives a rigorous treatment of entanglement measures in the general context of quantum field theory. It covers a broad range of models and the use of fields allows us to properly take the localization of systems into account. The required mathematical techniques are introduced in a self-contained way.

## Nonnegative Matrix And Tensor Factorizations

**Author :**Andrzej Cichocki

**ISBN :**0470747285

**Genre :**Science

**File Size :**76. 15 MB

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This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMF’s various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors’ own recently developed techniques in the subject area. Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms. Provides a comparative analysis of the different methods in order to identify approximation error and complexity. Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.

## The Geometry Of Special Relativity

**Author :**Tevian Dray

**ISBN :**9781466510470

**Genre :**Mathematics

**File Size :**71. 56 MB

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The Geometry of Special Relativity provides an introduction to special relativity that encourages readers to see beyond the formulas to the deeper geometric structure. The text treats the geometry of hyperbolas as the key to understanding special relativity. This approach replaces the ubiquitous γ symbol of most standard treatments with the appropriate hyperbolic trigonometric functions. In most cases, this not only simplifies the appearance of the formulas, but also emphasizes their geometric content in such a way as to make them almost obvious. Furthermore, many important relations, including the famous relativistic addition formula for velocities, follow directly from the appropriate trigonometric addition formulas. The book first describes the basic physics of special relativity to set the stage for the geometric treatment that follows. It then reviews properties of ordinary two-dimensional Euclidean space, expressed in terms of the usual circular trigonometric functions, before presenting a similar treatment of two-dimensional Minkowski space, expressed in terms of hyperbolic trigonometric functions. After covering special relativity again from the geometric point of view, the text discusses standard paradoxes, applications to relativistic mechanics, the relativistic unification of electricity and magnetism, and further steps leading to Einstein’s general theory of relativity. The book also briefly describes the further steps leading to Einstein’s general theory of relativity and then explores applications of hyperbola geometry to non-Euclidean geometry and calculus, including a geometric construction of the derivatives of trigonometric functions and the exponential function.

## The Proper Generalized Decomposition For Advanced Numerical Simulations

**Author :**Francisco Chinesta

**ISBN :**9783319028651

**Genre :**Technology & Engineering

**File Size :**33. 14 MB

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Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom. Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab© software used to generate these examples.

## Regularity Theory For Mean Field Game Systems

**Author :**Diogo A. Gomes

**ISBN :**9783319389349

**Genre :**Mathematics

**File Size :**20. 13 MB

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Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields.

## Schwinger S Quantum Action Principle

**Author :**Kimball A. Milton

**ISBN :**9783319201283

**Genre :**Science

**File Size :**32. 87 MB

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Starting from the earlier notions of stationary action principles, these tutorial notes shows how Schwinger’s Quantum Action Principle descended from Dirac’s formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. Part I brings out in more detail the connection between the two formulations, and applications are discussed. Then, the Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.