In my course on particle physics the KG and Dirac equations were explained in an historical way. However, it was more confusing than clarifying. The problems with the KG equation were
The probability density can take negative values.
Energy doesn't have a lower bound.
To solve this the Dirac equation was introduced. However, it only solves the first problem but not the second. Then the antiparticle interpretation is presented to solve the negative energies problem and everybody is happy. Also, the probability density problem of the KG equation is then solved saying that it is actually a charge current so it can take negative values...
So, if density is actually charge density and negative energies are antiparticles, what was the problem with the KG equation in the first place? How can we interpret the probability density of the Dirac equation as charge density if its always positive? Does QFT solve the negative energy problem without the need of postulating weird stuff like particles going backwards in time?
