Editing The Classical Theory of Fields (Book)
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 24: | Line 24: | ||
Finally, Yang-Mills by Atiyah. These lecture notes introduce some of the early results of the theory like the construction of instanton solutions and the conversion of these equations to Penrose's complex-geometry Twistor setting in a notation familiar to physicists, in coordinates, matrices, quaternions. As you will see in studying the mathematical aspects of Yang-Mills theory (YM) on 4-dimensional space-time, this dimension is particularly interesting. Penrose's book on Spinors and Space-Time shows this uniqueness in another way, converting the tensor calculus of GR into spinor calculus, another equally geometric formulation but specific to this dimension. He shows that the equations of both EM and GR can be put in this setting, and in his second volume expands on this to his Twistor-space formulation of particle physics. Milnor's Characteristic Classes and Atiyah's K-theory books pick up where Bott & Tu left off and characterize the algebra of vector bundles in two different ways. Characteristic classes are the language by which curvature and an invariant polynomial on a Lie group are associated to an underlying class of a vector bundle, which is the hinted combination of curvature and differential forms from earlier. This is used to count a deformation-invariant type of gauge theoretic state, and is known as a topological quantum number in quantum field theory and condensed matter physics. K-theory as another fundamental invariant looks at vector bundles with a fixed type of geometric structure but over all dimensions, meaning e.g. it describes properties of the rotation or unitary groups that are 'stable' in all dimensions. It has a subtle meaning, but is connected to spinors and Clifford algebras via Bott periodicity. | Finally, Yang-Mills by Atiyah. These lecture notes introduce some of the early results of the theory like the construction of instanton solutions and the conversion of these equations to Penrose's complex-geometry Twistor setting in a notation familiar to physicists, in coordinates, matrices, quaternions. As you will see in studying the mathematical aspects of Yang-Mills theory (YM) on 4-dimensional space-time, this dimension is particularly interesting. Penrose's book on Spinors and Space-Time shows this uniqueness in another way, converting the tensor calculus of GR into spinor calculus, another equally geometric formulation but specific to this dimension. He shows that the equations of both EM and GR can be put in this setting, and in his second volume expands on this to his Twistor-space formulation of particle physics. Milnor's Characteristic Classes and Atiyah's K-theory books pick up where Bott & Tu left off and characterize the algebra of vector bundles in two different ways. Characteristic classes are the language by which curvature and an invariant polynomial on a Lie group are associated to an underlying class of a vector bundle, which is the hinted combination of curvature and differential forms from earlier. This is used to count a deformation-invariant type of gauge theoretic state, and is known as a topological quantum number in quantum field theory and condensed matter physics. K-theory as another fundamental invariant looks at vector bundles with a fixed type of geometric structure but over all dimensions, meaning e.g. it describes properties of the rotation or unitary groups that are 'stable' in all dimensions. It has a subtle meaning, but is connected to spinors and Clifford algebras via Bott periodicity. | ||
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 | 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. 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. 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. | 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. | ||