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The Classical Theory of Fields (Book): Difference between revisions
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Kobayashi & Nomizu's book series is like a sequel to Sternberg's lectures in that its focus is on the definitions of geometric structures rather than algebraic invariants, and begins on connection theory where Sternberg's book ends. It also describes characteristic classes in an alternate way, via the Chern-Weil homomorphism with its starting point being curvature and connections rather than abstract algebraic invariants (since Milnor only references the curvature story in his appendix). Here, we can also get an impression of how gauge theory began in mathematics independently of its current incarnation in particle physics with Ehresmann, Weyl, and Cartan. Lie groups are the simplest curved spaces without the unnecessary restriction to two dimensions (there are no nonabelian 2-dimensional Lie groups), and their quotients offer alternative definitions of important topological spaces such as spheres, Grassmannians, Euclidean and Hyperbolic spaces. Continuing with Sternberg's explanation of the structure of Euclidean spaces, all of these 'symmetric spaces' and homogeneous spaces come with some naturally-equipped structures such as invariant 1-forms or metrics which are generalized to other compact Lie groups and used to give the most concrete constructions of curvature or other functions available in differential geometry. These spaces then became the starting point for more general bundle theory, as they are also equipped with natural bundles. Unlike these spaces, curvature in GR is not generally uniform and dynamically varies with the matter content of space-time. As such, Gauge theory is also equipped with the language to describe more general curvatures on manifolds. | Kobayashi & Nomizu's book series is like a sequel to Sternberg's lectures in that its focus is on the definitions of geometric structures rather than algebraic invariants, and begins on connection theory where Sternberg's book ends. It also describes characteristic classes in an alternate way, via the Chern-Weil homomorphism with its starting point being curvature and connections rather than abstract algebraic invariants (since Milnor only references the curvature story in his appendix). Here, we can also get an impression of how gauge theory began in mathematics independently of its current incarnation in particle physics with Ehresmann, Weyl, and Cartan. Lie groups are the simplest curved spaces without the unnecessary restriction to two dimensions (there are no nonabelian 2-dimensional Lie groups), and their quotients offer alternative definitions of important topological spaces such as spheres, Grassmannians, Euclidean and Hyperbolic spaces. Continuing with Sternberg's explanation of the structure of Euclidean spaces, all of these 'symmetric spaces' and homogeneous spaces come with some naturally-equipped structures such as invariant 1-forms or metrics which are generalized to other compact Lie groups and used to give the most concrete constructions of curvature or other functions available in differential geometry. These spaces then became the starting point for more general bundle theory, as they are also equipped with natural bundles. Unlike these spaces, curvature in GR is not generally uniform and dynamically varies with the matter content of space-time. As such, Gauge theory is also equipped with the language to describe more general curvatures on manifolds. | ||
Spin geometry specializes back to Riemannian geometry. | Spin geometry specializes back to Riemannian geometry. Earlier it was covered in multiple books that structured vector bundles (e.g. the Riemannian tangent bundle) have associated Lie-group fibered principal bundles to which the curvature can be transported. Now, we can take advantage of the topological (really, homotopical) structure of the special orthogonal group and turn the SO(n) bundle into a Spin(n) bundle. Converting this back into an associated vector bundle requires a representation of Spin(n), which we do not automatically have unlike SO(n). Calculating such possible representations involves the Clifford algebra of each orthogonal group, and when taken back to the vector bundle setting gives us a new differential operator - the Dirac operator on spinor sections - which squares modulo scalar curvature to the Riemannian Laplacian. The squaring to recover the Riemannian objects doesn't stop there, in some dimensions spinor quadratic maps can be constructed that turn spinor sections back into vector fields, realizing the sentence 'the square root of geometry.' This is a bit of a roundabout story, but it leads to new topological invariants which were not apparent from just the vector fields and Riemannian curvature, and new proofs of physical theorems like the positive energy conjecture of GR in an arguably more natural way. It is also worth noting that like Sternberg, the first chapter is spent on multilinear (Clifford) algebra and is a useful reference for its tables alone. | ||
The final basic text on gauge theory here, by [[Robert Hermann]], is rather special because he was the person to originally point out that the underlying geometry of the forces of the standard model was Gauge theory. He was often not credited, but those in the know knew it was him such as Singer or Gilkey. His book also adds substance, because it is the only one here that directly discusses models of physical phenomena and includes his personal thoughts on the direction of the field. All of his books are a must-read, for those pursuing research in an independent manner following our culture here. Manin's book on the other hand, represents his standing as one of the last Russian mathematical physicists deeply involved in both subjects. It takes the construction of instantons, the YM to Twistor conversion to a higher level (which is the complex geometry part), and the supermanifold enhancement of YM. We believe supersymmetry is heavily misinterpreted/mis-instantiated in physics, and hopefully you will see here that the direction mathematics takes it is different and useful. | |||
The most difficult book here by far is Besse's, as it assumes dexterity with nearly all of the above constructions and geometry. In short, an Einstein structure is the GR analog of an instanton, and this book constructions some and describes their topological consequences. | |||
=== Applications === | === Applications === | ||