Editing C-symmetry (section)

=== In geometry ===
For general [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s, one has a [[tangent bundle]], a [[cotangent bundle]] and a [[metric tensor|metric]] that ties the two together. There are several interesting things one can do, when presented with this situation. One is that the smooth structure allows [[differential equation]]s to be posed on the manifold; the [[tangent space|tangent]] and [[cotangent space]]s provide enough structure to perform [[differentiable manifold|calculus on manifolds]]. Of key interest is the [[Laplacian]], and, with a constant term, what amounts to the Klein–Gordon operator. Cotangent bundles, by their basic construction, are always [[symplectic manifold]]s. Symplectic manifolds have [[canonical coordinate]]s <math>x,p</math> interpreted as position and momentum, obeying [[canonical commutation relation]]s. This provides the core infrastructure to extend duality, and thus charge conjugation, to this general setting. 

A second interesting thing one can do is to construct a [[spin structure]]. Perhaps the most remarkable thing about this is that it is a very recognizable generalization to a <math>(p,q)</math>-dimensional pseudo-Riemannian manifold of the conventional physics concept of [[spinor]]s living on a (1,3)-dimensional [[Minkowski spacetime]]. The construction passes through a complexified [[Clifford algebra]] to build a [[Clifford bundle]] and a [[spin manifold]]. At the end of this construction, one obtains a system that is remarkably familiar, if one is already acquainted with Dirac spinors and the Dirac equation. Several analogies pass through to this general case. First, the [[spinor]]s are the [[Weyl spinor]]s, and they come in complex-conjugate pairs. They are naturally anti-commuting (this follows from the Clifford algebra), which is exactly what one wants to make contact with the [[Pauli exclusion principle]]. Another is the existence of a [[chiral symmetry|chiral element]], analogous to the [[gamma matrix]] <math>\gamma_5</math> which sorts these spinors into left and right-handed subspaces. The complexification is a key ingredient, and it provides "electromagnetism" in this generalized setting. The spinor bundle doesn't "just" transform under the [[pseudo-orthogonal group]] <math>SO(p,q)</math>, the generalization of the [[Lorentz group]] <math>SO(1,3)</math>, but under a bigger group, the complexified [[spin group]] <math>\mathrm{Spin}^\mathbb{C}(p,q).</math>  It is bigger in that it is a [[cover (mathematics)|double covering]] of <math>SO(p,q)\times U(1).</math>  

The <math>U(1)</math> piece can be identified with electromagnetism in several different ways. One way is that the [[Dirac operator]]s on the spin manifold, when squared, contain a piece <math>F=dA</math> with <math>A</math> arising from that part of the connection associated with the <math>U(1)</math> piece. This is entirely analogous to what happens when one squares the ordinary Dirac equation in ordinary Minkowski spacetime. A second hint is that this <math>U(1)</math> piece is associated with the [[determinant bundle]] of the spin structure, effectively tying together the left and right-handed spinors through complex conjugation.

What remains is to work through the discrete symmetries of the above construction. There are several that appear to generalize [[P-symmetry]] and [[T-symmetry]]. Identifying the <math>p</math> dimensions with time, and the <math>q</math> dimensions with space, one can reverse the tangent vectors in the <math>p</math> dimensional subspace to get time reversal, and flipping the direction of the <math>q</math> dimensions corresponds to parity. The C-symmetry can be identified with the reflection on the line bundle.  To tie all of these together into a knot, one finally has the concept of [[Transposition (mathematics)|transposition]], in that elements of the Clifford algebra can be written in reversed (transposed) order.  The net result is that not only do the conventional physics ideas of fields pass over to the general Riemannian setting, but also the ideas of the discrete symmetries.

There are two ways to react to this. One is to treat it as an interesting curiosity. The other is to realize that, in low dimensions (in low-dimensional spacetime) there are many "accidental" isomorphisms between various [[Lie group]]s and other assorted structures. Being able to examine them in a general setting disentangles these relationships, exposing more clearly "where things come from".