Interactions terms among vector fields in Lagrangian In the QED Lagrangian we have terms that look like $\partial_{\nu} A_{\mu}$ and $J_{\nu} A_{\mu}$.  We cannot have $A_{\mu} A^{\mu}$ as its coefficient would give a mass term, and for the photon we need $m=0$. 
We need to constrain ourselves to terms of mass dimension lower or equal to $4$ to ensure renormalisability. 
Is there any mathematical or physical reason (e.g. would lead to unstable field theory) that prevents terms like $A^{\nu} A^{\mu}\partial_{\nu} A_{\mu}$ or $A^4$?
Peskin and Schröder at some point days that 

Sensible theories of this type are difficult to construct because of the negative-norm states produced by the time dependent component $A^0$ of the vector field operator

and I’m trying to find examples to understand this statement, or alternative explanations.
 A: The reason these terms are omitted is they violate gauge invariance. This is problematic since, generically, a local field theory that doesn't preserve gauge invariance will lead to unitarity violation. One way to see this is that the $A_0$ component has a negative norm (as mentioned above). 
This is easy enough to show:
\begin{align}  \left| A _0  ( x )  \left| 0 \right\rangle \right| ^2 & = \left\langle 0 \right| A _0 ( x ) A _0 ( x ) \left| 0\right\rangle \\ 
& = \lim _{ x   \rightarrow y   } \left\langle 0 \right|  T \left[ A _0 ( x ) A _0 ( y ) \right] \left| 0\right\rangle \\ 
& = \lim _{ x \rightarrow y  } \int \frac{ d^4k }{ (2\pi)^4 } e ^{ - i k \cdot ( x - y )  } \frac{ - i g _{ 00 } }{ k ^2} \\ 
& = \int \frac{ d^3 k }{ (2\pi)^3 } \frac{1}{ 2 \left| {\mathbf{k}} \right| }( -  g _{ 00 } )  
\end{align} 
where in the limit I've taken $x_0>y_0$.
This quantity is clearly negative since $g_{00} = 1$ however its positive for the spatial components and hence they have positive norm. Imposing gauge invariance allows you to remove these states in a controlled (Lorentz invariant) way from the theory. 
