C-symmetry

 ( Quantum Field Theory ) 

In physics, charge conjugation is a transformation that switches all with their corresponding , thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symmetry is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are P-symmetry (parity) and T-symmetry (time reversal).

These discrete symmetries, C, P and T, are symmetries of the equations that describe the known fundamental forces of nature: electromagnetism, gravity, the strong and the . Verifying whether some given mathematical equation correctly models nature requires giving physical interpretation not only to continuous symmetries, such as motion in time, but also to its discrete symmetries, and then determining whether nature adheres to these symmetries. Unlike the continuous symmetries, the interpretation of the discrete symmetries is a bit more intellectually demanding and confusing. An early surprise appeared in the 1950s, when Chien Shiung Wu demonstrated that the weak interaction violated P-symmetry. For several decades, it appeared that the combined symmetry CP was preserved, until CP violation interactions were discovered. Both discoveries led to .

The C-symmetry is particularly troublesome, physically, as the universe is primarily filled with matter, not anti-matter, whereas the naive C-symmetry of the physical laws suggests that there should be equal amounts of both. It is currently believed that CP-violation during the early universe can account for the "excess" matter, although the debate is not settled. Earlier textbooks on cosmology, predating the 1970s, routinely suggested that perhaps distant galaxies were made entirely of anti-matter, thus maintaining a net balance of zero in the universe.

This article focuses on exposing and articulating the C-symmetry of various important equations and theoretical systems, including the Dirac equation and the structure of quantum field theory. The various fundamental particles can be classified according to behavior under charge conjugation; this is described in the article on C-parity.

$\backslash psi(x)\; =\; \backslash int\; d^3p\; \backslash sum\_\{\backslash sigma,n\}$

   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)
     

where $\backslash vec\; p$ is the momentum, $\backslash sigma$ is a spin label, $n$ is an auxiliary label for other states in the system. The $a$ and $a^\backslash dagger$ are creation and annihilation operators () and $u,\; v$ 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 relations, in that the one operator "undoes" what the other "creates". This implies that any given solution $u\backslash left(\backslash vec\; p,\; \backslash sigma,\; n\backslash right)$ must be paired with its "anti-solution" $v\backslash left(\backslash vec\; p,\; \backslash sigma,\; n\backslash right)$ 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 $u\backslash left(\backslash vec\; p\backslash right)$ is associated with a $v\backslash left(\backslash vec\; p\backslash right)$ 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 $u$ to be "dual to" some other solution $v,$ 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 representations $\backslash mathbf\{3\}$ and $\backslash overline\backslash mathbf\{3\}$ 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 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 invariants of the universal enveloping algebra for those symmetries. This is the case for both the Lorentz symmetry of the underlying spacetime 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 $u\backslash left(\backslash vec\; p,\; \backslash sigma,\; n\backslash right)$ correspond to particles, and solutions $v\backslash left(\backslash vec\; p,\; \backslash sigma,\; n\backslash right)$ 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.

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 $(p,q)$-dimensional pseudo-Riemannian manifold of the conventional physics concept of 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 are the , 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, analogous to the gamma matrix $\backslash gamma\_5$ 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 $SO(p,q)$, the generalization of the Lorentz group $SO(1,3)$, but under a bigger group, the complexified spin group $\backslash mathrm\{Spin\}^\backslash mathbb\{C\}(p,q).$ It is bigger in that it is a double covering of $SO(p,q)\backslash times\; U(1).$

The $U(1)$ piece can be identified with electromagnetism in several different ways. One way is that the on the spin manifold, when squared, contain a piece $F=dA$ with $A$ arising from that part of the connection associated with the $U(1)$ 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 $U(1)$ 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 $p$ dimensions with time, and the $q$ dimensions with space, one can reverse the tangent vectors in the $p$ dimensional subspace to get time reversal, and flipping the direction of the $q$ 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, 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 and other assorted structures. Being able to examine them in a general setting disentangles these relationships, exposing more clearly "where things come from".

$(i\{\backslash partial\backslash !\backslash !\backslash !\backslash big\; /\}\; -\; q\{A\backslash !\backslash !\backslash !\backslash big\; /\}\; -\; m)\; \backslash psi\; =\; 0$

One wishes to find a charge-conjugate solution

$(i\{\backslash partial\backslash !\backslash !\backslash !\backslash big\; /\}\; +\; q\{A\backslash !\backslash !\backslash !\backslash big\; /\}\; -\; m)\; \backslash psi^c\; =\; 0$

A handful of algebraic manipulations are sufficient to obtain the second from the first. Standard expositions of the Dirac equation demonstrate a conjugate field $\backslash overline\backslash psi\; =\; \backslash psi^\backslash dagger\backslash gamma^0,$ interpreted as an anti-particle field, satisfying the complex-transposed Dirac equation

$\backslash overline\backslash psi(-i\{\backslash partial\backslash !\backslash !\backslash !\backslash big\; /\}\; -\; q\{A\backslash !\backslash !\backslash !\backslash big\; /\}\; -\; m)\; =\; 0$

Note that some but not all of the signs have flipped. Transposing this back again gives almost the desired form, provided that one can find a 4×4 matrix $C$ that transposes the gamma matrices to insert the required sign-change:

$C^\{-1\}\backslash gamma\_\backslash mu\; C\; =\; -\backslash gamma\_\backslash mu^\backslash textsf\{T\}$

The charge conjugate solution is then given by the involution

$\backslash psi\; \backslash mapsto\; \backslash psi^c=\backslash eta\_c\backslash ,\; C\backslash overline\backslash psi^\backslash textsf\{T\}$

The 4×4 matrix $C,$ called the charge conjugation matrix, has an explicit form given in the article on gamma matrices. Curiously, this form is not representation-independent, but depends on the specific matrix representation chosen for the gamma group (the subgroup of the Clifford algebra capturing the algebraic properties of the gamma matrices). This matrix is representation dependent due to a subtle interplay involving the complexification of the spin group describing the Lorentz covariance of charged particles. The complex number $\backslash eta\_c$ is an arbitrary phase factor $|\backslash eta\_c|=1,$ generally taken to be $\backslash eta\_c=1.$

Conventionally, $\backslash gamma\_5$ is used as the chirality operator. Under charge conjugation, it transforms as

$C\backslash gamma\_5\; C^\{-1\}\; =\; \backslash gamma\_5^\backslash textsf\{T\}$

and whether or not $\backslash gamma\_5^\backslash textsf\{T\}$ equals $\backslash gamma\_5$ depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have that $\backslash gamma\_5^\backslash textsf\{T\}\; =\; \backslash gamma\_5$, while $\backslash gamma\_5^\backslash textsf\{T\}\; =\; -\backslash gamma\_5$ is obtained in the Majorana basis. A worked example follows.

   = \left( \psi^\dagger \gamma^0 \right)^\textsf{T}
   = \begin{pmatrix} 0 & I \\ I & 0\end{pmatrix} \begin{pmatrix} \psi_\text{L}^* \\ \psi_\text{R}^* \end{pmatrix}
   = \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
     

   = \begin{pmatrix} \psi_\text{L}^c\\ \psi_\text{R}^c \end{pmatrix}
   = \eta_c C \overline\psi^\textsf{T}
   = \eta_c \begin{pmatrix} -i\sigma^2 & 0 \\ 0 & i\sigma^2\end{pmatrix} \begin{pmatrix} \psi_\text{R}^* \\ \psi_\text{L}^* \end{pmatrix}
   = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{R}^* \\ i\sigma^2\psi_\text{L}^* \end{pmatrix}
     

    =  \begin{pmatrix} \psi_\text{L}^{cp} \left(t, \vec x\right)\\ \psi_\text{R}^{cp}\left(t,\vec x\right) \end{pmatrix}
    = \eta_c \begin{pmatrix} -i\sigma^2\psi_\text{L}^*\left(t, -\vec x\right) \\ i\sigma^2\psi_\text{R}^*\left(t, -\vec x\right) \end{pmatrix}