- #1
nonequilibrium
- 1,439
- 2
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?
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?