Derivations of the Klein-Gordon equation such as the one given by Phoenix in here, are based on studying the wave equation of the wave function of a relativistic particle. In this case, the Klein-Gordon equation has the simple interpretation of describing a wave function whose support is in the hyperboloid of a given mass. I was however wondering if there was more of a "field theoretic" derivation of this equation. Although I am still looking for a group theoretic approach, I would like a derivation that:
- Avoids mentioning the quantization of a relativistic particle. I am interested in a point of view where the main objective is to describe the quantum behaviour of a field. If this later has a particle interpretation or not should not be important at this stage.
- Avoids quantum considerations for that matter. The Klein-Gordon equation describes a perfectly sensible classical field theory.
- Is based on relativistic covariance of the theory.
- Is based on a good notion of what a free theory is.
I hope that this approach elucidates facts such as that both the free scalar and Dirac fields satisfy this equation. Even the electromagnetic potential does (in the Lorenz gauge) in the absence of sources. Thus, it seems reasonable that any free Poincaré covariant field should satisfy this equation (modulo gauge invariance). Moreover, I hope that this elucidates the role that the representation the field is in plays in the refinements of the equation of motion. By refinements I mean the Dirac equation or the additional constraint imposed by the Lorenz gauge.