It looks like solutions of the KG eqn travel faster than light, because if $$\omega^2 - k^2 = m^2$$ then $$\mid\ \omega\mid \ > \ \mid k\ \mid$$ and I thought the wave velocity was $\omega / k$.
How do I interpret this?
I heard something about particles following the packet velocity, which isn't too fast, but why can't these fast wave fronts be used for signalling irrespective of what any packets are doing?
Edit:
In the meantime I've been looking at propagators but it all looks too grandiose to me. To my eyes, the KG equation is nothing but a guitar string where each point on the string is pulled back to the quiescent position by a spring. I could set up a simulation of that and ping it with a tall, narrow gaussian. Sure enough, the waves would go faster with the extra springs because it's all stiffer, and that effect would be stronger at long wavelengths where the springs would dominate the curvature, so we'd have a $d\omega/dk$. But I don't think anything would hit infinity (like this propagator thinks) and I'd be very surprised if the far end of the string stayed completely motionless until the packet velocity arrived (or would it?)
So why all this talk of complex numbers and contour integrals that were never part of the original problem? Even if I promoted the displacement to a complex number, the real and imaginary parts would form two independent systems because all the diffs are second order so there's nothing to mix them up. What am I missing?
The possible dupe doesn't answer this. It tells us what wave packets are but not why the phase velocity can't carry a signal.
No comments:
Post a Comment