Ryder in his QFT book writes in eqn (2.20):
Probability density, $\rho = \frac{i\hbar}{2m}(\phi^*\frac{\partial \phi}{\partial t} – \phi \frac{\partial \phi^*}{\partial t})$
Then in the next paragraph he writes: Since the Klein-Gordon equation is second order, $\phi$ and $\frac{\partial \phi}{\partial t}$ can be fixed arbitrarily a given time, so $\rho$ may take on negative values,..
What does he mean by this line?
Best Answer
He means that since it is a second order differential equation, it is completely determined by two sets of information, namely the value of $\phi$ and ${\dot \phi}$ at some time $t=t_0$. In other words, $\phi(t_0)$ and ${\dot \phi}(t_0)$ paramaterizes the solution. We can therefore choose them to take any values, some which will imply a negative value of $\rho$.