We can derive the Klein-Gordon equation using the relativistic energy-momentum relation, and here, I have no problem, it is an easy thing to do. However, I found by research that it applies only to spinless, spin 0, particles, but why is that. How can I deduce this just by looking at the equation.
Basically my question generalizes to: "how can I deduce, which kind of particles a particular wave equation describes, by looking at the equation?", since I would also like to know why I can say that the Dirac equation describes massive spin-1/2 particles and that Einstein's field equations describes a massless spin-2 field.
Best Answer
My total guess, which I hope other answers can correct is that, if you remember back to the Schroedinger Equation, it was based on the classical, non-relativistic version of energy. It does not incorporate spin and is not Lorentz invariant.
The Klein Gordon equation does take into account SR, by using E$^2$ = $p^2c^2 + m^2c^2$, but it can be interpreted as a scalar field, producing spin 0 particles, because it transforms as a scalar field.
The glib answer is: don't just look, follow through on finding a solution(s) and when the second quantization is performed, you will know if you have a vector field, which produces particles with spin.