Gauge Fields, Knots and Gravity (Knots and Everything) by John Baez (PDF)

Description

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell’s equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein’s equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.

User’s Reviews

Editorial Reviews: From Scientific American “The book is clearly written and should be accessible to readers who have a good undergraduate preparation in mathematics or physics. Each part of the book ends with a list of references that will enable the reader to pursue the material presented in greater detail.” Review “This book is a great introduction to many of the modern ideas of mathematical physics including differential geometry, group theory, knot theory and topology. It uses as ‘physical excuses’ to introduce these topics Maxwell theory, Yang-Mills theories and general relativity (including its Ashtekar reformulation). The level of the book is gauged to advanced physics/math undergraduates and graduate students. The style of the book is quite lively and explanations are very clear. The treatment is mathematically and physically self-contained … I would strongly recommend this nicely written book for anyone interested in teaching the contemporary ideas of mathematical physics to an audience of physicists (especially if that audience is interested in particle physics/gravity). It offers an excellent way of treating the subject with mathematical rigor while keeping the physical motivation and usefulness of these mathematical concepts close at hand. For the individual reader, it is a great way to be lured into the study of the mathematics that underlies contemporary theoretical physics.”

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As stated in every other review, this book is simply phenomenal. Not to parrot the rest of them, I’d like to point out perhaps it’s only flaw, in my opinion.The author abuses notation quite uncomfortably. As a physicist trying to learn math, at times the author mixes the mathematicians need for rigor with the physicists sloppy shorthand. And it leaves me utterly baffled. When we are defining the standard flat connection on sections as smooth whatevers over a local trivialization, I find that leaving the fact that we pulled back two things and pushed forward two others to be confusing. I tend to get lost trying to break down the steps. I’ll look at a definition and break it down piecewise and associate different facts about the object to other definitions. And then I’ll think I screwed up because something is missing, but in reality this being pushed forward was “trivial.”Had quite a few occurrences where I try to understand everything from a mathematicians point of view and get lost only to realize that he was just being sloppy like a physicist would. I think a little more consistency would have been great. If he is going to belabor pull backs in one section and then ignore then in the next, it needs to be clearly stated.Also, every single notation used involved parenthesis. A(v) = v(A,v(a))= (Ava(A())V). Really, there are a dozen different delimiters that we could have used. Why does everything have to be enclosed in parenthesis?Aside from these two nitpicks, the book up there with the best I’ve ever used.

⭐I was surprised that I didn’t come across this book until recently. And it was a very pleasant one, must I add.It’s a delight to read this book. Even by reading the introductions to all the chapters an ‘uninitiated’ explorer can actually get a good idea of how these exotic areas of mathematics are interconnected and can be used to describe physical realities. There is a great deal of aestehtics in this interplay, and the book brings them out in its true spirit.The authors are rigorous, but definitely not so much as some of the standard texts in these areas. And they admitted it in the preface itself. The book is like an ‘invitation’ to this extraordinarily beautiful and modern area of mathematical physics.I am a great fan of John Baez. He has been writing an excelling series of blogs (TWF, and then Azimuth) for a long time, and that IMO probably ranks into the class of the finest ever science writings. No wonder the book carries the strain of the same genius, that of explaining the abstruest in simplied manner, in abundance.

⭐This is a fantastic book!! John Baez is one of the best writers that I have ever seen. He just has this way of of being so straightforward…….I bought this book because I am a topologist and I have always had a hard time understanding differential forms and deRham cohomology. This book is so good at giving that subject a physical and geometrical interpretation. Just about every other book I read on those subjects seems to get so algebraic and I always ask myself “Where did the geometry go?”, but not here. This book could serve as a great introduction to riemannian geometry. The physical interpretations of this high powered math is so tastefully done. I am very satisfied with this book!

⭐This gives a concise and lucid exposition of the subject.

⭐Amazing book. The authors are very straight forward and not afraid to tell us about some of the short comings that some the models in physics have inherent to them. This is not a bad thing in the slightest. In fact, it is an indicator that there is some much that we still need to learn as a species.

⭐Compralo ya!Es el mejor libro para entender cada concepto importante en Topologia Differencial y como se aplican a la fisica teorica clasica. Y por supuesto, no podrias entender jamas la teoria de strings o la Gravedad Quantica ,a menos q entiendas una gran cantidad de Topologia Differencial . Estos profesores, son tambien Fisicos de vanguardia…ellos entienden bien lo que hace falta y como ilustrarlo con ejemplos de la fisica clasica antes de aplicarlos a la nueva fisica.Desde las manifolds ( quien entiende eso desde la primera vez?) ,pasando por los bundles y cohomologias , hasta las formas de Chern!!!No pierdas tiempo : Compralo ya!!!!Very well written, especially the section on Chern-Simon Theory.

⭐still reading it is tough

⭐Es un libro perfecto para matemátic@s, especialmente con experiencia en geometría diferencial (nivel de post-grado), que quieran saben cómo se traduce lo que están haciendo/estudiando en la realidad física. El lenguaje que utiliza es eminentemente matemático, por lo que seguramente sea más útil para matemátic@s que quieran entender sus aplicaciones físicas que para físic@s que quieran profundizar en la dimensión matemática que hay debajo de sus descripciones de la realidad.Prosa increíble llena de explicaciones profundamente clarificadoras. Se entretiene muchos más en generar una intución sólida que en demostraciones técnicas. Incluso con conceptos matemáticos y teorías que creía que ya había entendido en profundidad, el enfoque del libro me ha servido para darme cuenta de aspectos que no había entendido del todo o en los que aún no había reparado.De los mejores libros técnicos que he leído nunca.This book is a very good book for intermediate student and advance students. Also this book contains lots of pictures which is aslo helpful .

⭐Thos book is incredibly clear, and is perfect for physicists looking to learn some modern mathematics. There are hardly any rigorous proofs in this book, however the author gives such clear descriptions and motivations that they are not needed. The exercises are also great for learning the material.

⭐Ottimo libroThis book explains brilliantly, but briefly, topics in Mathematical Physics, and is a worthy buy if one has Some basic prerequisite knowledge on Differential Geometry and topology.

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