Let's consider the solutions $\psi$ of the Klein-Gordon equation:
$$\bigg{(}\frac{\partial^2}{\partial t^2}-\Delta + m^{2}\bigg{)}\psi(x) = 0$$
and define:
$$\rho = \frac{i}{2m}\bigg{(}\psi^{\*}\frac{\partial \psi}{\partial t}-\frac{\partial \psi^{\*}}{\partial t}\psi\bigg{)} \quad \mbox{and} \quad {\bf{j}} = -\frac{i}{2m}(\psi^{\*}\nabla\psi -\nabla \psi^{\*}\psi)$$
Then, with a little algebra one can show that the following continuity equation is satisfied:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot {\bf{j}} = 0$$

**Question:** I've heard that because the above continuity equation has both positive and negative solutions, the solutions of the Klein-Gordon equation do not have a probabilistic interpretation. Why is that? More precisely, what is the connection between probabilistic interpretations and continuity equations? Does it have something to do with Noether's Theorem?