Editing C-symmetry (section)

==In general settings==
The analog of charge conjugation can be defined for [[higher-dimensional gamma matrices]], with an explicit construction for Weyl spinors given in the article on [[Weyl–Brauer matrices]]. Note, however, spinors as defined abstractly in the representation theory of [[Clifford algebra]]s are not fields; rather, they should be thought of as existing on a zero-dimensional spacetime. 

The analog of [[T-symmetry]] follows from <math>\gamma^1\gamma^3</math> as the T-conjugation operator for Dirac spinors. Spinors also have an inherent [[P-symmetry]], obtained by reversing the direction of all of the basis vectors of the [[Clifford algebra]] from which the spinors are constructed. The relationship to the P and T symmetries for a fermion field on a spacetime manifold are a bit subtle, but can be roughly characterized as follows. When a spinor is constructed via a Clifford algebra, the construction requires a vector space on which to build. By convention, this vector space is the [[tangent space]] of the spacetime manifold at a given, fixed spacetime point (a single fiber in the [[tangent manifold]]). P and T operations applied to the spacetime manifold can then be understood as also flipping the coordinates of the tangent space as well; thus, the two are glued together. Flipping the parity or the direction of time in one also flips it in the other.  This is a convention. One can become unglued by failing to propagate this connection.

This is done by taking the tangent space as a [[vector space]], extending it to a [[tensor algebra]], and then using an [[inner product]] on the vector space to define a [[Clifford algebra]]. Treating each such algebra as a fiber, one obtains a [[fiber bundle]] called the [[Clifford bundle]].  Under a change of basis of the tangent space, elements of the Clifford algebra transform according to the [[spin group]]. Building a [[principle fiber bundle]] with the spin group as the fiber results in a [[spin structure]].

All that is missing in the above paragraphs are the [[spinor]]s themselves. These require the "complexification" of the tangent manifold: tensoring it with the complex plane. Once this is done, the [[Weyl spinor]]s can be constructed. These have the form
:<math>w_j = \frac{1}{\sqrt{2}}\left(e_{2j} - ie_{2j+1}\right)</math>

where the <math>e_j</math> are the basis vectors for the vector space <math>V=T_pM</math>, the tangent space at point <math>p\in M</math> in the spacetime manifold <math>M.</math> The Weyl spinors, together with their complex conjugates span the tangent space, in the sense that
:<math>V \otimes \mathbb{C} = W\oplus \overline W</math>

The alternating algebra <math>\wedge W</math> is called the [[spinor space]], it is where the spinors live, as well as products of spinors (thus, objects with higher spin values, including vectors and tensors).

<!---

::{{font color|grey|Taking a break; this section should expand on the following statements: }}

::{{font color|grey| Obstruction to building spin structures is [[Stiefel–Whitney class]] {{math|w}}{{sub|2}} }}
::{{font color|grey| Complex conjugation exchanges the two spinors }}
::{{font color|grey| [[Dirac operator]]s may be defined that square to the Laplacian i.e. the square of the Levi-Civita connection (plus scalar curvature plus line bundle curvature) }}
::{{font color|grey| the curvature of the line bundle is explicitly {{math|F {{=}} dA}} ergo it must be E&M }}

--->