I am going through Schwartz's "Quantum Field Theory and the Standard Model". When trying to construct the Lagrangian for QED on chapter 8.2.2 his approach is as follows:
We want a theory where the four degrees of freedom of a 4-vector are expressed as three degrees of freedom mixed together plus one extra. This would be $4=3\oplus 1$. I understand why this has to be the case. He then proposes a first attempt at building a Lagrangian as $$\mathcal{L}=-\frac{1}{2}\partial_\nu A_\mu\partial_\mu A_\nu + \frac{1}{2}m^2A_\mu^2.\tag{8.17}$$ From this you can get that the equations of motion are given by $$(\Box + m^2)A_\mu = 0.\tag{8.18}$$ So far so good. But then he states that we can see from this, that this is the Lagrangian for four scalar fields instead of the lagrangian from a massive spin-1 field. So in this case $4=1\oplus 1\oplus 1\oplus 1. $ This is where I get lost. Where can we actually see that from the equation of motion?
He then goes about constructing a better Lagrangian and ends up with $$\mathcal{L}=\frac{a}{2}A_\mu\Box A_\mu + \frac{b}{2}A_\mu\partial_\mu\partial_\nu A_\nu + \frac{1}{2}m^2A_\mu^2,\tag{8.20}$$ which he claims, has the correct property of being $4=3\oplus 1$.
I would like to get some clarity about the way we can know these things from looking at the equation.
