If two real wave functions and separately satisfy a Klein-Gordon equation with the same mass , then the separate Klein-Gordon equations can be replaced by one Klein-Gordon equation for a complex wave function:

(4.50) |

Using the definition of charge in equation 4.49, we see that and have opposite charge and that a real wave function has zero charge. In general we have

complex scalar fields are charged, |

real scalar fields are uncharged. |

We now examine the charge for a superposition of positive-energy and superposition of negative-energy solutions to the Klein-Gordon equation. If is a positive-energy solution to the Klein-Gordon equation with momentum , the superposition of all such positive-energy solutions is

(4.51) |

where is a weighting function of three-momentum only and is given by equation 4.19. The charge for this general positive-energy solution is

(4.52) |

where the last step follows because implies .

For a superposition of negative-energy solutions, . Thus specifies a particle with positive charge and a particle with the same mass but negative charge.

For zero charge spin-0 particles, the wave function must be real: . For a single momentum we can write

(4.53) |

From the relativistic wave equation for spin-0 particles and the interpretation of its wave functions, we are lead to three solutions which correspond to the electric charges , and for every momentum . The relativistic quantum theory thus reveals the charge degree of freedom of particles.

According to our current understanding, there are no fundamental spin-0 particles. However, if we do not probe the internal structure of mesons they can be considered as Klein-Gordon particles. For the pion system with the approximation that each state has the same mass, we write (by convention)

(4.54) |

By convention it is the positive-charged boson which is the particle and the negative-charge fermion which is the particle.

2004-03-18