Series on Knots and Everything - Vol. 4
GAUGE FIELDS, KNOTS AND GRAVITY
by John Baez & Javier P Muniain (University of California, Riverside)
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.
Contents:
- Electromagnetism:
- Maxwell's Equations
- Manifolds
-
Vector Fields
- Differential Forms
- Rewriting Maxwell's Equations
- DeRham Theory in Electromagnetism
- Gauge Fields:
- Symmetry
- Bundles and Connections
- Curvature and the Yang-Mills Equation
- Chern-Simons Theory
- Link Invariants from Gauge Theory
- Gravity:
- Semi-Riemannian Geometry
- Einstein's Equation
- Lagrangians for General Relativity
- The ADM Formalism
- The New Variables
Readership: Mathematicians, mathematical physicists and theoretical
physicists.
"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."
| Classical & Quantum Gravity |
"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."
| 480pp |
Pub. date: Oct 1994 |
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