# Homework Help: Klein-Gordon signal propagation

1. Oct 19, 2007

### alle.fabbri

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

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

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted