Why is $4=3\oplus 1$? What are propagating modes? Etc
In <PERSON>'s QFT book, he said that the vector representation of the Lorentz group, $V_\mu$ that is four-dimensional, is the direct sum of two irreducible representations of $SO(3)$: a spin-0 representation, which is one-dimensional, and a spin-1 representation, which is three-dimensional. i.e. $4=3\oplus 1$.
Question 1:
Is there an explicit reason why so? I am looking for both a mathematical reason that is also physical. 656
The book just indicated it without stating exactly how.
Is it because the 0th component of the vector is the spin-0 and the other components are spin-1? If so, this doesn't make sense to me since the vector field $A_\mu$ itself is spin-1.
The book then continues to guess a Lagrangian for a massive spin-1 field and came up with $$ \mathcal{L}=-\frac{1}{2}\partial_\nu A_\mu\partial_\mu A_\nu+\frac{1}{2}m^2A^2_\mu $$ Which has the equations of motion $$ (\partial^2+m^2)A_\mu=0 $$ And then says that it has "4 propagating modes" and $A_\mu$ is actually 4 scalar fields and therefore the Lagrangian is not for a vector field. In this case this is $4=1\oplus 1\oplus 1\oplus 1$.
Question 2:
What does he mean by propagating modes? Are the fields just scalar because the equation of motion is for each index of $A_\mu$ or is there a much better meaning of why?
He never actually defined "Propagating modes" at all. Anyways, the book continues with another guess which is the Proca Lagrangian $$ L = \frac{a}{2}A^\mu\partial^2A_\mu+\frac{b}{2}A_\mu\partial_mu\partial_\nu A_\nu +\frac{1}{2}m^2A_\mu A^\mu $$ Where it says that the $\partial_\mu A_\mu$ contraction forces $A_\mu$ to be a 4-vector since if each component $A_\mu$ transforms as scalar then $\partial_\mu A_\mu$ is not Lorentz invariant.
Question 3:
I seem to not understand what the book meant by that. Why is it not Lorentz Invariant?
Thank you 656