5 edition of Topology, geometry, and gauge fields found in the catalog.
Includes bibliographical references (p. 401-405) and index.
|Statement||Gregory L. Naber|
|Series||Applied mathematical sciences -- 141|
|LC Classifications||QC20.7.T65 N34 2011|
|The Physical Object|
|Pagination||xii, 419 p. ;|
|Number of Pages||419|
|ISBN 10||9781441978943, 9781441978950|
|LC Control Number||2011293269|
The development ofnewcourses is a natural consequence of a high levelof excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Naber combines a knowledge of his subject with an excellent informal writing style. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics TAM. The simplest such group is U 1which appears in the modern formulation of quantum electrodynamics QED via its use of complex numbers. The mathematical discipline of topology is introduced in an intuitive way and is then combined with the methods of differential geometry to show how the emergence of gapless states can be understood. Two such mathematical configurations are equivalent describe the same physical situation if they are related by a transformation of this abstract coordinate basis a change of local section, or gauge transformation.
The coordinate transformation has affected both the coordinate system used to identify the location of the measurement and the basis in which its value is expressed. He gives full details to all difficult calculations and shows good judgment in deciding what is difficult versus what is not. The reader is assumed to have a minimal understanding of what an electromagnetic field is, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, and a solid background in real analysis and linear algebra with some of the vocabulary of modern algebra. In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point a local section of the fiber bundle and express the values of the objects of the theory usually " fields " in the physicist's sense using this basis.
He has undertaken the arduous task of researching the broad field with its extensive literature, learning the material himself, class testing it in lectures, and agonizing over the best ways to present it. The mathematical discipline of topology is introduced in an intuitive way and is then combined with the methods of differential geometry to show how the emergence of gapless states can be understood. This book is warmly recommended to specialists in mathematics and physics, and especially to PhD students interested in the topic. This second edition of the book includes a new chapter on singular homology theory and a new appendix outlining Donaldson's beautiful application of gauge theory to the topology of compact, simply connected, smooth 4-manifolds with definite intersection form. Reviews of the first edition: "It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics
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The goal is to weave together rudimentary notions from the classical gauge theories of physics and the topological and geometrical concepts that became the mathematical models of these notions.
Preface In Egypt, geometry was created to measure the land. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the finite-dimensional Lie group, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber over that point.
When analyzing the dynamics of a gauge geometry, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. Determination of the likelihood of possible measurement outcomes proceed by: establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information establishing a probability distribution of measurement outcomes for each possible physical situation convolving these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information These assumptions have enough validity across a wide range of energy scales and experimental conditions to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life: geometry, heat, and electricity, eclipses, spaceflight, etc.
In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point a local section of the fiber bundle and express the values of the objects of the theory usually " fields " in the physicist's sense using this basis. I particularly enjoyed Chapter 2 where many fundamental physical examples are discussed at great length in a reader friendly fashion.
Newton created the calculus to study the motion of physical objects apples, planets, etc. For someone who plans to compute right along with the examples, this book is a must-buy. Quantum field theories[ edit ] Other than these classical continuum field theories, the most widely known gauge theories are quantum field theoriesincluding quantum electrodynamics and the Standard Model of elementary particle physics.
Naber combines a knowledge of his subject with an excellent informal writing style. In terms of its ability to teach a subject to the novice, this book ranks right up there with many classics Read more. The topics covered in this book can be found in many other sources, but the present volume discusses with great care those aspects and notions which are particularly important in gauge and gauge fields book.
Global symmetry[ geometry ] In physicsthe mathematical description of any physical situation usually contains excess degrees of freedom ; the same physical situation is equally well described by many equivalent mathematical configurations.
Local symmetry[ edit ] Use of fiber bundles to describe local symmetries[ edit ] In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time.
Naber favors examples that deal with concrete spaces and revisits them whenever appropriate As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity.
Moreover, the presentation is backed by many useful and relevant examples and I am convinced that any beginner in gauge theory will find them very useful. A symbol and a subject index are included. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics TAM.
Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a gauge fixing prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group the global symmetry group, or perhaps even the trivial group.
TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied mathe matical Sciences AMS series, which will focus on advanced textbooks and research level monographs.
I particularly enjoyed Chapter 2 where many fundamental physical examples are discussed at great length in a reader friendly fashion.
All of these are then developed rigorously in the remaining chapters. He has undertaken the arduous task of researching the broad field with its extensive literature, learning the material himself, class testing it in lectures, and agonizing over the best ways to present it.
Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. Practical ways of carrying out topological calculations are described. Classical gauge theory[ edit ] Classical electromagnetism[ edit ] Historically, the first example of gauge symmetry discovered was classical electromagnetism.
A gauge transformation whose parameter is not a constant function is referred to as a local symmetry ; its effect on expressions that involve a derivative is qualitatively different from that on expressions that don't. As long as this transformation is performed globally affecting the coordinate basis in the same way at every pointthe effect on values that represent the rate of change of some quantity along some path in space and time as it passes through point P is the same as the effect on values that are truly local to P.
Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. This second edition of the book includes, in an Appendix, a much expanded sketch of Seiberg-Witten gauge theory, including a brief discussion of its origins in physics and its implications for topology.
The techniques of calculation in a continuum theory implicitly assume that: given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these bounds fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.MATHEMATICS STUDENT ORGANIZATION.
DREXEL UNIVERSITY. MAY 15, Gauge Fields and Geometry. A Picture Book. Greg Naber. Abstract. Mathematicians are a strange breed. The delight they take in an idea very often appears to be in direct proportion to what everyone else in the world finds off-putting about it; they revel in.
Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields Gerd Rudolph, Matthias Schmidt. Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge atlasbowling.com first building block.
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Topology, Geometry and Gauge fields: Foundations Gregory L. Naber. This is a book on topology and geometry, and like any book on subjects as vast as these, it has a point of view that guided the selection of topics.
The author’s point of view is that the rekindled interest that mathematics and physics have shown in each other of late should.
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