Klein-Gordon signal propagation

  • #1
Given Klein-Gordon equation for a particle of mass m in covariant notation
[tex] \left[ \partial_{\mu} \partial^{\mu} + \frac{m_0^2 c^2}{\hbar^2} \right] \phi = 0[/tex]
show that the solution preserves causality, i.e. signals have a velocity not higher than c.

HINT: You can build up a quantity [tex] j^{\mu} = j^{\mu} \left( \phi \right)[/tex] which satisfy a continuity equation [tex]\partial_{\mu} j^{\mu} = 0[/tex]. Then apply the Gauss Theorem to the lower layer of a light cone [tex]x_{\mu} x^{\mu} = 0[/tex] of finite height [tex]\overline{x^0}[/tex] delimited by two 3-sphere of radium R, lower, and [tex]R - \overline{x^0}[/tex], upper, and show that the space integral of [tex] j^0[/tex] can only decrease during the evolution.

I have built up a 4-vector that behaves like that but I can't realize how this justify the fact that Klein-Gordon is a causal theory.

Any ideas?
 

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