Editing C-symmetry (section)

=== 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.