Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? 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.
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Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. 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 c This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. 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. Get A Copy. Paperback , pages. More Details Original Title.
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Sep 06, Jon Paprocki rated it really liked it. This text is aimed at undergraduates, but I would readily recommend it to grad students as well. It is a really fast and easy introduction to the topics mentioned in the title.
I only skimmed the more basics sections on topics like fiber bundles, connections, etc. I also didn't read the section on quantum gravity yet since I'm not very interested in it. But as far as an introduction into gauge theory, Chern-Simons theory, and applications of quantum field theory to knots and 3-manifold invariants, a grad student in either physics or math could do worse than spending a few days going through these sections.
The writing is not overly dense, the exercises are well thought out and usually easy, and the authors easily switch back and forth between mathematical and physical language. While I had wished that some topics had more written on them when do you ever not? The lost star is just because I can't see myself ever opening the book again now that I've read it once unless I decide to read the quantum gravity section. It is very pedagogical and won't be a very useful reference to have around.
My ideal textbook is somehow both pedagogical and an excellent reference, an almost impossible feat to achieve. May 08, David rated it really liked it Shelves: physics , mathematics. I read this little book some years ago and I really enjoyed. It shows you all the mathematical bricks you need to know in order to build your understanding some of the most advanced theories in Physics.
The explanations are clear and the style is one that physicists will certainly appreciate also mathematicians, but perhaps they would prefer more theorem and proof style, absent here. It can be read by advanced undergraduates, as it assumes nothing more than advanced calculus and linear algebra I read this little book some years ago and I really enjoyed. It can be read by advanced undergraduates, as it assumes nothing more than advanced calculus and linear algebra, but as it is usually said, some "mathematical maturity" would also help.
The book begins discussing the math tools needed for advanced Physics, basically differential geometry and group theory, and it does it in a very fast way, so don't expect anything about the Gauss-Bonnet theorem or discussion of exotic groups SU 5 or even more exotic ones. The example of the electromagnetism, rewritting the Maxwell equations in the language of differential forms is very well done.
The explanation of the concept of fiber bundle is also brilliant, taking into consideration that it's a difficult concept. There's a section of knot theory that it wasn't specially interesting for me but that it's very well explained, beginning with the basics.
It seems that knots could be involved in the unification of General Relativity and Quantum Mechanics. The final section of the book is about Einstein's relativity, perhaps the most beautiful theory of Physics, trying to link it to what it has been showed previously: a geometric theory that can be described with Cartan's formalism. Curious point: all the groups involved in the electromagnetic, weak and strong forces are compacts, and when a group is compact is equivalent to a sum of irreducible representations not proved in the text, and probably an advanced proof, but we physicists have faith.
The irreducible representations are the particles. The exception is General Relativity, that it is not compact. A change of a topological property that creates a really big problem and literally divides physics in two. Overall, a very nice book if you're serious about learning advanced theories in Physics but you're also a beginner.
I see many good reviews here, so let me offer a different perspective. I don't know for whom this book was intended. According to the author, for mathematicians who want to learn some physics, or vice versa. But I don't see anybody benefiting from this book really. Physicists who haven't learned mathematics somewhere else won't learn it here. Sure they may remember some definitions and concepts, but this book is simply too superficial. You won't learn a mathematical concept if there aren't all the I see many good reviews here, so let me offer a different perspective.
You won't learn a mathematical concept if there aren't all the necessary details presented to you. And if you don't learn it properly, intuition you think you have about the concept might be wrong. Not to mention that you are incompetent in making conclusions and do practical calculations with understanding.
Studying this book, you will not learn about manifolds, bundles etc. There is no way around a math textbook in topology and differential geometry. Handwaving only helps if you know the material.
Sure, some mathematical concepts had a neat geometric interpretation. But, atlhough the book is "mathematical" funny , most mathematics isn't introduced well. As my teacher from linear algebra said: "That's the way physicists like it: as many indices as possible". I haven't read the last part, but just skimming through it I see it's not very different from other GR books regarding mathematical style.
The book will have you thinking that explanations are very intuitive, by using friendly and unformal language, while many mathematical concepts are introduced very non elegantly for example, tensor products and conncections. I didn't get past the Yang-Mills equation, so I can't speak for the rest , which does not contribute to understanding. Most mathematicians didn't study any quantum mechanics and have studied very little electrodynamics, so this book won't help them gain any physical insight.
For example, quantum mechanics is explained in less than three pages in just four pages the author introduced the transition amplitude, action, lagrangian and a wavefunction , followed by the discussion on Aharonov-Bohm effect, and very superficial for that matter. Instead, author chooses to focus on wormholes and other mumbo jumbo with no physical relevance.
So, in order to benefit from this book, you have to already know physics and mathematics presented? Well, what is the point then in reading this book? I'm a physics student and I learned some differential geometry covered in Loring Tu's An Introduction to Manifolds and Differential Geometry and this book was simply a waste of time.
It is true that this makes me biased, I simply can't know how much would an undergraduate benefit from this book. But I stand by what I said: reading books such as this is either useless or decieving. They cover much material, lure you with pictures, informal language and bad jokes and leave you not with knowledge but with pictures and handwaving explanations. But the thing that annoyed me the most are the childlike excitment about every mathematical equation and the citations.
This book has more citations than a theology textbook while we're at it, it very much resembles a theology textbook. One of my favourite is "The boundary of a boundary is zero. Really, J. Wheeler is the one who said that? Then he continues with the usual habit physics textbooks have of deforming the history by saying that deRham invented differential forms. The citation that I was thinking about while reading this book is: "A student's knowledge is no larger than the set of examples thoroughly understood.
I would say that I learned more about gauge theory just by reading the preface of Gauge Theory and Variational Principles David Bleecker , a book I highly recommend I read only half of it, but it was very enjoyable. Oct 10, Saman rated it really liked it.
Gauge Fields, Knots and Gravity