Radio waves from an antenna using the propagator?

[nuclearhead]

I want to find a simple example of using the Feynman propagator: 1/(|x-x'|²-(t-t')²)
and also to show that no signals an travel faster than light.
So I was thinking about waves emitted from an antenna.

Tell me if I got this right?

Assume a static source in space at x=0 and varying in time: δ(x')exp(iEt') (maybe an electron vibrating in a wire?)

Then integrating over all time for the source

∫1/(|x-x'|²-(t-t')²)δ(x')exp(iEt') dt = ( exp(iEt - i|E||x| ) /|x|

This gives the amplitude for finding a photon at x,t I think? The probability is 1/|x|² which agrees with the inverse square law for waves.

Is this valid to use the propagator of a virtual particle in this case or should you use the delta function δ(x²-t²) for a "real" photon?

Now assume another static source this time the antenna is transmitting a more complicated wave δ(x')f(t'). Is it possible to show that this wave form gets transmitted to the point (x,t)? Does the Feynman propagator alter this waveform? Also, can it be shown that no information from this waveform travel faster than light?

 

[vanhees71]

Well, for a radio wave radiated from an antenna must not use the Feynman propagator but the retarded propagator. It's very important to distinguish the different kinds of propagators, depending on the application. Also in linear-response theory in quantum field theory you get the correct result, i.e., the retarded propagator for your application. Have a look at

http://fias.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

for some details of the real-time formalism of (relativistic) quantum many-body theory.