Editing C-symmetry (section)

==Informal overview==
Charge conjugation occurs as a symmetry in three different but closely related settings: a symmetry of the (classical, non-quantized) solutions of several notable differential equations, including the [[Klein–Gordon equation]] and the [[Dirac equation]], a symmetry of the corresponding quantum fields, and in a general setting, a symmetry in (pseudo-)[[Riemannian geometry]]. In all three cases, the symmetry is ultimately revealed to be a symmetry under [[complex conjugation]], although exactly what is being conjugated where can be at times obfuscated, depending on notation, coordinate choices and other factors.

=== In classical fields ===
The charge conjugation symmetry is interpreted as that of [[electrical charge]], because in all three cases (classical, quantum and geometry), one can construct [[Noether current]]s that resemble those of [[classical electrodynamics]]. This arises because electrodynamics itself, via [[Maxwell's equations]], can be interpreted as a structure on a [[U(1)]] [[fiber bundle]], the so-called [[circle bundle]]. This provides a geometric interpretation of electromagnetism: the [[electromagnetic potential]] <math>A_\mu</math> is interpreted as the [[Connection (mathematics)|gauge connection]] (the [[Ehresmann connection]]) on the circle bundle. This geometric interpretation then allows (literally almost) anything possessing a complex-number-valued structure to be coupled to the electromagnetic field, provided that this coupling is done in a [[gauge-invariant]] way. Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion. More formally, one says that the equations must be gauge invariant under a change of local [[coordinate frame]]s on the circle. For U(1), this is just the statement that the system is invariant under multiplication by a phase factor <math>e^{i\phi(x)}</math> that depends on the (space-time) coordinate <math>x.</math>  In this geometric setting, charge conjugation can be understood as the discrete symmetry <math>z = (x + iy) \mapsto \overline z = (x - iy)</math> that performs complex conjugation, that reverses the sense of direction around the circle.

=== In quantum theory ===
In [[quantum field theory]], charge conjugation can be understood as the exchange of [[particle]]s with [[anti-particle]]s.  To understand this statement, one must have a minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via [[perturbation theory]]. A key ingredient to this process is the [[quantum field]], one for each of the (free, uncoupled) differential equations in the system. A quantum field is conventionally written as
:<math>\psi(x) = \int d^3p \sum_{\sigma,n} 
    e^{-ip\cdot x} a\left(\vec p, \sigma, n\right) u\left(\vec p, \sigma, n\right) + 
    e^{ip\cdot x} a^\dagger\left(\vec p, \sigma, n\right) v\left(\vec p, \sigma, n\right)
</math>

where <math>\vec p</math> is the momentum, <math>\sigma</math> is a spin label, <math>n</math> is an auxiliary label for other states in the system. The <math>a</math> and <math>a^\dagger</math> are [[creation and annihilation operators]] ([[ladder operator]]s) and <math>u, v</math> are solutions to the (free, non-interacting, uncoupled) differential equation in question. The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions. However, via perturbation theory, approximate solutions can be constructed as combinations of the free-field solutions. To perform this construction, one has to be able to extract and work with any one given free-field solution, on-demand, when required. The quantum field provides exactly this: it enumerates all possible free-field solutions in a vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators. 

The creation and annihilation operators obey the [[canonical commutation relation]]s, in that the one operator "undoes" what the other "creates". This implies that any given solution <math>u\left(\vec p, \sigma, n\right)</math> must be paired with its "anti-solution" <math>v\left(\vec p, \sigma, n\right)</math> so that one undoes or cancels out the other. The pairing is to be performed so that all symmetries are preserved. As one is generally interested in [[Lorentz invariance]], the quantum field contains an integral over all possible Lorentz coordinate frames, written above as an integral over all possible momenta (it is an integral over the fiber of the [[frame bundle]]). The pairing requires that a given <math>u\left(\vec p\right)</math> is associated with a <math>v\left(\vec p\right)</math> of the opposite momentum and energy. The quantum field is also a sum over all possible spin states; the dual pairing again matching opposite spins. Likewise for any other quantum numbers, these are also paired as opposites. There is a technical difficulty in carrying out this dual pairing: one must describe what it means for some given solution <math>u</math> to be "dual to" some other solution <math>v,</math> and to describe it in such a way that it remains consistently dual when integrating over the fiber of the frame bundle, when integrating (summing) over the fiber that describes the spin, and when integrating (summing) over any other fibers that occur in the theory. 

When the fiber to be integrated over is the U(1) fiber of electromagnetism, the dual pairing is such that the direction (orientation) on the fiber is reversed. When the fiber to be integrated over is the SU(3) fiber of the [[color charge]], the dual pairing again reverses orientation. This "just works" for SU(3) because it has two dual [[fundamental representation]]s <math>\mathbf{3}</math> and <math>\overline\mathbf{3}</math> which can be naturally paired. This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite [[charge (physics)|charges]] in the fully abstract sense. In physics, a charge is associated with a generator of a continuous symmetry. Different charges are associated with different eigenspaces of the [[Casimir invariant]]s of the [[universal enveloping algebra]] for those symmetries. This is the case for ''both'' the Lorentz symmetry of the underlying [[spacetime]] [[manifold (mathematics)|manifold]], ''as well as'' the symmetries of any fibers in the fiber bundle posed above the spacetime manifold. Duality replaces the generator of the symmetry with minus the generator. Charge conjugation is thus associated with reflection along the [[line bundle]] or [[determinant bundle]] of the space of symmetries.

The above then is a sketch of the general idea of a quantum field in quantum field theory. The physical interpretation is that solutions <math>u\left(\vec p, \sigma, n\right)</math> correspond to particles, and solutions <math>v\left(\vec p, \sigma, n\right)</math> correspond to antiparticles, and so charge conjugation is a pairing of the two. This sketch also provides enough hints to indicate what charge conjugation might look like in a general geometric setting. There is no particular forced requirement to use perturbation theory, to construct quantum fields that will act as middle-men in a perturbative expansion. Charge conjugation can be given a general setting.

=== 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".