Renormalizing solutions of the Klein-Gordon equation

[nonequilibrium]

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 \varphi(\mathbf r,t) is a solution of the KG equation, then we can define a new field as \psi(\mathbf r,t) := \varphi(\mathbf r,t) / ||\varphi(t)|| where ||\phi(t)|| := \int_{\mathbb R^3} \varphi(\mathbf r',t) \mathrm d \mathbf r'.

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?

 

[nonequilibrium]

Actually I suppose one wouldn't expect the norm to be Lorentz-invariant, so the resulting equation is probably not relativistic.

 

[Demystifier]

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.
http://xxx.lanl.gov/abs/1205.1992
Sec. 8.3.1, especially Eqs. (8.39)-(8.40).