I have just been introduced to the Klein-Gordon Equation and the Dirac Equation for the first time. The way they were explained to me, these equations govern the (relativistic) evolution of spin-0 and spin-1/2 particles respectively.
My question is, what fundamentally differentiates particles with different spins, and how can those differences have such large effects on the system's dynamics that particles with different spins are described by two fundamentally different equations of motion?
Even though I am still an undergrad, I understand PDEs well enough to know that even slightly altering a system's equation of motion can lead to dramatically different physical solutions. So the fact that spin is able to entirely change the relevant EOMs is quite surprising to me, and seems to suggest that it plays a central role in these systems that I do not yet fully appreciate.