Charge Conjugation

Consider a positive-energy solution with charge $e$. The equivalent Klein-Gordon equation satisfied by the positive-energy solution is

$$i\hbar\partial_t\phi_e^{(+)} = \sqrt{m^2c^4 + (-i\hbar c\vec{\nabla} - e\vec{A})^2}\ \phi_e^{(+)} + eA_0\phi_e^{(+)} .$$ (4.77)

The negative energy solution satisfies

$$i\hbar\partial_t\phi_e^{(-)} = -\sqrt{m^2c^4 + (-i\hbar c\vec{\nabla} - e\vec{A})^2}\ \phi_e^{(-)} + eA_0\phi_e^{(-)} .$$ (4.78)

Taking the complex conjugate of the negative energy equation gives

$$-i\hbar\partial_t{\phi_e^{(-)}}^* = -\sqrt{m^2c^4 + ( i\hbar c\vec{\nabla} - e\vec{A})^2}\ {\phi_e^{(-)}}^* + eA_0{\phi_e^{(-)}}^* ,$$

$$i\hbar\partial_t{\phi_e^{(-)}}^* = \sqrt{m^2c^4 + (-i\hbar c\vec{\nabla} + e\vec{A})^2}\ {\phi_e^{(-)}}^* - eA_0{\phi_e^{(-)}}^*.$$ (4.79)

Comparing equation 4.77 with equation 4.79 shows

$$\fbox{$\displaystyle {\phi_{-e}^{(-)}}^* \propto \phi_e^{(+)}$}\ .$$ (4.80)

Thus ${\phi_{-e}^{(-)}}^*$ is the charge conjugate solution and represents the charge conjugate state of $\phi_e^{(+)}$. Similarly, ${\phi_e^{(+)}}^*$ is the charge conjugate state of $\phi_{-e}^{(-)}$. If we (arbitrarily) call the particle described by $\phi_e^{(+)}$ ``the particle'', then we call the particle described by ${\phi_{-e}^{(-)}}^*$ the antiparticle. For example, if we call the $\pi^+$ meson the particle, then the $\pi^-$ meson is the antiparticle. The undesirable negative-energy solutions have now been interpreted as antiparticles.

Neutral particles fit into this picture too, in that the charge-conjugate state is the state itself. In other words, neutral particles are their own antiparticles. Let $\phi\equiv\phi_e^{(+)}$ and $\phi_C\equiv{\phi_{-e}^{(-)}}^*$ so that we can write

$$\phi_C = \alpha\phi,$$ (4.81)

where $\alpha$ is a proportionality constant which has to be real. $\alpha$ can be deduced since for neutral particles both $\phi$ and $\alpha\phi$ are real. Realizing that

$$(\phi_C)_C = \phi$$ (4.82)

it follows that

$$(\alpha\phi)_C = \alpha^2\phi = \phi$$ (4.83)

so that $\alpha^2 = 1$ and $\alpha = \pm 1$. Accordingly there exist two different kinds of neutral particles, namely

1. neutral particles with positive charge parity, ie. $\alpha=+1$,

   
   

   $$\phi_C=\phi ,$$
   
   

2. neutral particles with negative charge parity, ie. $\alpha=-1$,

   
   

   $$\phi_C=-\phi .$$
   
   

Neutral particles are thus eigenfunctions of the charge conjugation operator, while charged particles are not.