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Why don't scalar fields propagate superluminally?

  1. Mar 11, 2013 #1
    This is a really basic question, but...

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

    [itex]\partial_t^2 \phi + (k^2 + m^2)\phi = 0.[/itex]

    This has solutions

    [itex]\phi \sim e^{\pm \sqrt{k^2 + m^2}t}[/itex]

    and the sound speed should be

    [itex]\omega_k/k = \sqrt{1 + m^2/k^2} \geq 1.[/itex]

    in which case perturbations of the scalar field propagate superluminally at all scales. This is clearly wrong, but why?
  2. jcsd
  3. Mar 11, 2013 #2
    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.
  4. Mar 12, 2013 #3
    Of course. Thanks a lot!
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