Why don't scalar fields propagate superluminally?

[ramparts]

This is a really basic question, but...

Say I have a massive scalar field obeying the Klein-Gordon equation linearized about flat space,

$\partial_t^2 \phi + (k^2 + m^2)\phi = 0.$

This has solutions

$\phi \sim e^{\pm \sqrt{k^2 + m^2}t}$

and the sound speed should be

$\omega_k/k = \sqrt{1 + m^2/k^2} \geq 1.$

in which case perturbations of the scalar field propagate superluminally at all scales. This is clearly wrong, but why?

 

[The_Duck]

I think it's the group velocity that shouldn't be superluminal. The group velocity is

$\frac{\partial w_k}{\partial k} = \frac{2k}{2 \sqrt{k^2 + m^2}} = \frac{1}{\sqrt{1 + m^2/k^2}} < 1$

unless $m = 0$ in which case the group velocity is 1, as it should be.

 

[ramparts]

Of course. Thanks a lot!