Hi, I was looking at the lagrangian and conserved currents for the free complex scalar field and it looks like it has a striking similarity to the conserved current for probability: [tex] \frac{\partial \rho}{\partial t}=\nabla\cdot \vec{j} [/tex] where [itex]j_i =-i(\psi^{\ast}\partial_i \psi - \psi\partial_i \psi^{\ast})[/itex] and [itex]\rho[/itex] is the probability density. Then with the action [tex] \mathcal{L}=\partial_\alpha \psi^{\ast}\partial^\alpha \psi [/tex] the conserved current is [tex] j^{\alpha}=-i(\psi^\ast \partial^\alpha \psi - \psi \partial^\alpha \psi^\ast ) [/tex] Then I had the thought that with the conservation of probability current, the above lagrangian appears to be a lagrangian for a free field of...probability. Now I'm aware that the complex scalar field is used to describe various spin-0 particles, but has anyone heard of any other possible thoughts on this lagrangian, maybe back when it was first put forward, or when anyone was just looking at relativistic quantum mechanics about 100 years ago?
There's no coincidence. The so-called Schroedinger field described by a complex wavefunction in a Galilei invariant space-time goes into the complex free KG field. But when the probability interpretation of the current derived from phase invariance fails, a new interpretation is necessary, the electric charge one. This is classic stuff described in a gazillion of books.