quantum field theory - the causality and the anti-particles

How can I quantitatively and qualitatively understand the fact that there is a relevence between the existence of anti-particles and the causality?

Answer

This is a consequence of the fact that there is no positive frequency function which is zero outside the light cone. If you have a particle in relativity, its dynamics require that it goes faster than light, and to restore causality, it must go back in time. This is explained in my answer here: Is anti-matter matter going backwards in time? .

If you have a quantum particle with positive energy, the propagation function $G(x-y)$ is the amplitude to go from x to y. This propagation is said to be causal if the propagator is zero unless x is to the future of y, so that in a time-space decomposition, $G(t,r)$ is zero for t<0. In this case, the Fourier transform $G(\omega,k)$ cannot vanish for all $\omega<0$, because it is impossible for a nonzero function and its Fourier transform to both be exactly zero in a half-plane.

To see this, the condition of vanishing of $G(t,r)$ for t<0 implies the analyticity of the Fourier transform for $\omega$ with a negative imaginary part, since in this region, the Fourier transform of G becomes a sum of decaying exponentials. An analytic function can't be zero in a region without being zero everywhere, so the Fourier transform of a future directed function is not strictly positive energy.

Because of this, there is no relativistic particle formalism in which the particles both have positive energies and causal propagation. You can either deal with fields, in which case the particle notion is non-local, or you can deal with particles, but then they go back in time.

The back-in-time formalism is using the standard non-causal Feynman propagator, which is

$$ G(\omega,k) = {i\over \omega^2 - k^2 - m^2 - i\epsilon}$$

up to numerator modifications for higher spin, with the $i\epsilon$ pole prescription. This has two poles in $\omega$ for any $k$, and the pole prescription pushes one pole to have slightly positive imaginary part and the other pole to the slightly negative imaginary part. There are singularities in both directions in imaginary $\omega$ direction, which means that the propagation is non-causal.

The part that goes forward in time is the positive energy part; the part that goes back in time is the negative energy part.