Relationship between the Klein-Gordon equation and Poincaré invariance 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.
 A: *

*The Lagrangian has to be constructed, demanding Lorentz invariance, symmetry properties, field content, and locality. The last, for example, tells you you must only keep a finite number of derivatives.

*The field content is what one means by the representation of the Poincare group the field lives in. This information is required to construct invariants that you can potentially put in your Lagrangian. For example, the invariant constructed would be different for a scalar $\phi$ or a spinor $\phi$.

*Suppose you want to look at spinless particles. Then, you look for fields that transform in the trivial representation of the Lorentz group(i.e. under the identity map), and thus are by definition what one would call scalar fields. The most general Lagrangian would look like(for a single field)- $$L=a(\partial\phi)^2+b(\partial\phi)^4+....+a'\phi+b'\phi^2+...$$

*You then consider the following physical criteria-(a) The equation of motion must be second order, because in a suitable limit it must carry the same information as Newton's second law, and that requires only 2 initial parameters must be free; and (b) If I wish to interpret these as wave equations, I see that including terms higher order than $\phi^2$ spoils the linearity; I will not be able to superpose my waves meaningfully.

*Your truncated Lagrangian is now a the Klein gordon Lagrangian, with a source term. It allows superposition of solutions to be a solution-and this is the hallmark of non-interacting. The waves 'pass right through each other'. If, for example, they scattered into different things, superposition would be broken. This "scattering" precisely encodes interaction, and to have no scattering(i.e. no interaction), you must truncate at quadratic order.

*Finally, for completeness, even though you don't necessarily want this, from a more EFT perspective- you start with the most general possible Lagrangian. You argue on grounds of renormalizability and keep only a finite number of terms. One can show that terms with higher dimensions would contribute in smaller and smaller amounts to the scattering process, and so on an appropriate scale, you can truncate things to order $4$ (that's the bar to decide what are the so called 'relevant' or marginally relevant operators). These marginally relevant operators have done the job of taking you away from a fixed point of the theory(where it is scale invariant), so you simply trace your steps back-and you find yourself at the massless Klein-Gordon Lagrangian. All other theories are deformations of this theory with relevant operators, the simplest being the mass term.
