Is the KleinGordon Eq the Wave Eq for Spin0 Scalar particles

[Trixie Mattel]

If we take the Lagrangian of a spin-0 scalar field and use the Euler-Lagrange equation, we end up with the Klein-Gordon equation. Does that mean that the wave equation of spin-0 scalar particles is the Klein-Gordon equation?Thank you

 

[stevendaryl]

If we take the Lagrangian of a spin-0 scalar field and use the Euler-Lagrange equation, we end up with the Klein-Gordon equation. Does that mean that the wave equation of spin-0 scalar particles is the Klein-Gordon equation?

There is a problem with interpreting the Klein-Gordon equation as describing the wave function for a particle, in the sense that the Schrödinger equation describes the wave function of a particle. The problem is that the conserved current associated with Klein-Gordon is not guaranteed to be positive. In the case of the Schrödinger equation, there is a conserved current that can be interpreted as the probability current for the particle (since probabilities have to be positive).