Classical Example
Under the charge conjugation (C for short), charge $q \to -q$, electric field $\mathbf E \to - \mathbf E$, but the solution of the dynamical equation, the position vector, remains invariant $\mathbf r(t) \to \mathbf r(t)$.
In the theories particle and antiparticle do not mix (e.g. Schroedinger's Equation, Pauli's Equation), charge symmetry is almost trivial.
Dirac Theory
Dirac equation, \[(i\partial_\mu \gamma^\mu - eA_\mu\gamma^\mu + m)\psi(x) = 0
\] describes the relativistic motion of electrons as well as positrons. $e = 1.602176565(35)\times 10^{-19} \mathrm{C}$ is just a positive number, so-called elementary charge. In general, the solution $\psi(x)$ contains both the electron state and the positron state. Here we take $g_{\mu\nu} = \mathrm{diag}\{-,+,+,+\}$.
In free particle theory where $A = 0$, there are four independent plane wave solutions: two positive energy solutions describe electrons $u_s(p) e^{+ip\cdot x}, \; s=\pm\frac{1}{2}$ and two negative energy solutions describe the positrons $v_s(p) e^{-ip\cdot x},\; s=\pm\frac{1}{2}$, $E_{\mathbf p} = \sqrt{\mathbf p^2+m^2}$. A general solution (a wave packet) is a superposition of the two pieces: $\psi(x) \equiv \psi^+(x) + \psi^-(x)$ where \[
\psi^+(x) = \sum_{s=\pm}\int \frac{\mathrm{d}^3p}{(2\pi)^32E_{\mathbf p}} b_s(\mathbf p) u_s(p) e^{+ip\cdot x}, \\
\psi^-(x) = \sum_{s=\pm}\int \frac{\mathrm{d}^3p}{(2\pi)^32E_{\mathbf p}} d^*_s(\mathbf p) v_s(p) e^{-ip\cdot x}. \] $b, d^*$ are some c-number smooth functions.
As a relativistic wave function approach, the charge conjugation would be implemented as a "unitary" spinor matrix: $C \bar C \triangleq C (\beta C^\dagger \beta) \equiv 1$. This matrix should transform a plane wave electron to a plane wave positron or vice versa, i.e. $C u_s(p)e^{+ip\cdot x} \sim v_s(p) e^{-ip\cdot x}$. We can immediately see that this is not possible if the charge conjugation spares the exponential function. We conclude that in Dirac theory, the charge conjugation must be implemented as an anti-unitary spinor operator. Thus, we require \[
C \left( u_s(p) e^{+ip\cdot x}\right) =\eta_c v^*_s(p) e^{-ip\cdot x} \\
C \left( v_s(p) e^{-ip\cdot x} \right) = \xi_c u^*_s(p) e^{+ip\cdot x} \] where $|\eta_c| = |\xi_c| = 1$. For simplicity, we'll take these constant phases to unity. The charge conjugated field is denoted as, \[
C\psi(x) \equiv \psi^c(x) = \sum_{s=\pm}\int \frac{\mathrm{d}^3p}{(2\pi)^32E_{\mathbf p}} \Big[ b^*_s(\mathbf p) v^*_s(p) e^{-ip\cdot x} + d_s(\mathbf p) u^*_s(p) e^{+ip\cdot x} \Big]
\]
It is conventional to define the unitary part of $C$ by a new spinor matrix $\mathcal C$ (curly C),
\[ C \psi(x) \triangleq \mathcal C \beta \psi^*(x) \equiv \mathcal C \bar\psi^T(x) \] (Here adding a $\beta$ is just a convention.). It is easy to see, $\mathcal C$ is unitary $\mathcal{C} \bar{\mathcal{C}}= 1$. An example of the choice of $\mathcal C$ is (Srednicki p. 242, 2007), \[
\mathcal C =
\begin{pmatrix}
0 & -1 & 0 & 0 \\
+1 & 0 & 0 & 0 \\
0 & 0 & 0 & +1 \\
0 & 0 & -1 & 0
\end{pmatrix}
\] The theory must be invariant under the charge conjugation, or $\mathcal L \overset{C}{\to} \mathcal L$, which implies $\mathcal C \gamma_\mu = - \gamma_\mu^T \mathcal C$. Applying to the $u,v$ spinors, $\mathcal C \bar u^T_s(p) = v_s(p), \mathcal C \bar v^T_s(p) = u_s(p)$.
Heuristically, the transformation can be viewed as a two-step procedure:
- swap the particle species, by exchanging $u_s(p)$ and $v_s(p)$ (and of course also the sign of the charge);
- reverse the time, by conjugating the exponential factor.
(i\partial_\mu \gamma^\mu {\color \red +} eA_\mu\gamma^\mu + m){\color \red {\psi^c(x)}} = 0
\] Note the sign change. It is easy to check that if $\psi(x)$ satisfies the original Dirac wave equation,
$\psi^c(x)$ satisfies this equation, which just confirms that the Dirac theory is invariant under the charge conjugation.
To compare with the classical charge conjugation, the dynamical equation (the equation of motion) is still invariant under the charge conjugation. But the solution of it would change under the charge conjugation, $\psi(x) \overset{C}{\to} \psi^c(x)$.
Quantum field theory
In quantum field theory, charge conjugation is implemented as a unitary operator, which we shall call $C$ (not be confused with the anti-unitary spinor operator introduced in the previous section). The definition is simple (We have also taken a particular choice of a possible phase factor. See Weinberg, 2005): \[C^{-1} b_s(p) C = d_s(p), \quad C^{-1} d_s(p) C = b_s(p), \quad C^{-1} a_\lambda(p) C = - a_\lambda(p), \] where $b_s(p), d_s(p), a_\lambda(p)$ are the electron, positron and photon annihilation operators, respectively.
For the free Dirac field, \[
\psi(x) = \sum_{s=\pm}\int \frac{\mathrm{d}^3p}{(2\pi)^32E_{\mathbf p}} \Big[ b_s(\mathbf p) u_s(p) e^{+ip\cdot x} + d^\dagger_s(\mathbf p) v_s(p) e^{-ip\cdot x} \Big].
\] It can be shown, after some algebra (e.g., Srenicki p.225) that for the quantum fields, \[
C^{-1} \psi(x) C = \mathcal C \bar\psi^T(x) \equiv \psi^c(x), \quad C^{-1} A^\mu(x) C = - A^\mu(x), \quad C^{-1}\varphi(x)C = \varphi^*(x). \] Then, it can be shown immediately that the Lagrangian is invariant under the charge conjugation.
Alot of notations are abused, although they are different quantities (quantum field theoretical vs. Dirac relativistic wavefunctional). Superficially, this resembles the Dirac equation result introduced in the previous section, although we should point out that here $\psi(x)$ is a field operator instead of a relativistic wave function. But this similarity means that we can do it one way or another. The same answer would be obtained. That's also the reason of the heavy abuse of notations.
To compare with the classical case and the Dirac case, the solution of the dynamical equation changes, $\psi(x) \overset{C}{\to} \psi^c(x)$. Notation-wise, this is similar to the Dirac case. See Table 1.
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| Table 1 |
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| Figure 1. $\langle p s \left| J^\mu(x) \right| p' s'\rangle \equiv e^{-i(p-p')\cdot x} \bar u_s(p)e \Gamma^\mu u_{s'}(p')$ |
A theory invariant under charge conjugation does not automatically implies a symmetric solution. The symmetry could be broken by either explicit theory (e.g. weak interaction, $\theta$-term etc.), or spontaneous symmetry breaking. In general, the charge distribution, the matrix element of the charge current operator between two physical states can be written as, \[
\langle p s \left| J^\mu(x) \right| p' s' \rangle \equiv e \bar u_s(p) \Gamma^\mu u_{s'}(p') \exp[-i(p-p')\cdot x].
\] Here $\Gamma^\mu = \Gamma^\mu(p-p')$ is some spinor operator. $\Gamma^\mu$ can be represented by the spinor basis, \[ \Gamma^\mu(q) = F_1(q^2) \gamma^\mu + F_2(q^2) \frac{i}{2m_e} \sigma^{\mu\nu} q_\nu + F_3(q^2) \frac{1}{2m_e}\gamma_5 \sigma^{\mu\nu} q_\nu,
\] here $q = p-p'$, $\sigma^{\mu\nu} = \frac{i}{2} [ \gamma^\mu, \gamma^\nu ]$. Under the charge conjugation,
$\gamma^\mu \overset{C}{\to} \mathcal C^{-1}(\gamma^\mu)^T \mathcal C = -\gamma^\mu$;
$\sigma^{\mu\nu} \overset{C}{\to} \mathcal C^{-1}(\sigma{^\mu\nu})^T \mathcal C = -\sigma^{\mu\nu}$;
$\gamma_5\sigma^{\mu\nu} \overset{C}{\to}\mathcal C^{-1}(\gamma_5\sigma{^\mu\nu})^T \mathcal C = -\gamma_5\sigma^{\mu\nu}$.
