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AdrianMay

- 121

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I've heard it said that the superluminal phase velocity of the KG eqn is not a problem for relativistic causality because signals travel at the packet/group velocity, which is the inverse of the phase velocity (c being 1). I'm a bit skeptical of this.

We can strip away all the quantum mystique and just consider the (1 dimensional) KG eqn to describe a guitar string with restoring springs along it, or (to put the same thing the other way around) a row of simple harmonic oscillators coupled together. Either way, we can bang on one end with a delta function and watch waves of all wavelengths spread along it at various superluminal speeds including infinity. To convince ourselves that signals can't travel at the phase velocity, we'll somehow have to prove that all those different components add up to zero at every event that's spacelike separated from the impulse. This would seem like a remarkable coincidence - there may be traveling nodes, but zero for the whole continuous spacetime region?

I'm not sure how to type formulae here, but I think I may have proved the opposite. We can consider an event at a spacelike interval from the impulse. Long waves travel faster under KG, so at that event, waves with k from 0 to K will be influencing psi at that event, but the shorter ones won't have arrived yet. We know K because we know that w^2 = k^2 + m^2. I'm not entirely sure what the amplitudes of the components of the Fourier transform of the delta function are when the velocity isn't trivial, but let's just make it a constant. Whatever, you wind up with an integral over k of an exponential of i times something complicated but real. It doesn't really matter that it's complicated. The answer will always be some complex number of a known non-zero magnitude.