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I just saw some lectures where they claim that the Klein Gordon equation is the lowest order equation which is Lorentz invariant for a scalar field.

But I could easily come up with a Lorentz invariant equation that is first order, e.g.

$$

(M^\mu\partial_\mu + m^2)\phi=0

$$

where M is a generic matrix.

Now, something should be wrong with this equation, because, as Dirac showed, if we want a first order equation the field needs to be a spinor.

But I don't clearly understand why this first order equation is not Lorentz invariant. I mean, $$M^\mu\partial_\mu$$ is a scalar, so the equation is invariant, isn't it?

Is it maybe because the matrix M changes form by changing reference system, so that we could find privileged systems (e.g. a reference where the matrix is diagonal)?