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Renormalizing solutions of the Klein-Gordon equation

  1. Jul 1, 2012 #1
    It is said that the solutions of the Klein-Gordon equation cannot be interpreted as probability densities since the norm isn't conserved in the time evolution.

    Now a pretty evident idea seems to be to renormalize the solution at each moment so that it is renormalized (and hence interpretable as a probability density) by definition... More exactly, if [itex]\varphi(\mathbf r,t)[/itex] is a solution of the KG equation, then we can define a new field as [itex]\psi(\mathbf r,t) := \varphi(\mathbf r,t) / ||\varphi(t)||[/itex] where [itex]||\phi(t)|| := \int_{\mathbb R^3} \varphi(\mathbf r',t) \mathrm d \mathbf r'[/itex].

    This defines a new norm-preserving field, which of course evolves according to a new equation (not KG). I'm not sure if this new equation is relativistic (i.e. Lorentz-invariant). I would think so, since I would expect the norm to be Lorentz-invariant, but I'm not entirely sure. Has the behaviour of this solution been investigated? Or is there an obvious reason why it is of no interest?
  2. jcsd
  3. Jul 1, 2012 #2
    Actually I suppose one wouldn't expect the norm to be Lorentz-invariant, so the resulting equation is probably not relativistic.
  4. Jul 2, 2012 #3


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    Mr. vodka, what you propose makes sense but is not new. It even can be viewed as being "covariant", provided that it is interpreted as CONDITIONAL probability. See e.g.
    Sec. 8.3.1, especially Eqs. (8.39)-(8.40).
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