Electromagnetic field quantization

[smallgirl]

I see what I did was wrong, and I can see that I have made the mistake several times. I guess I just don't spot it, but I am not at all confident with this...I had another go in the penultimate post, but I don't think the way I've permuted indices is correct..

 

[vela]

$$\frac{1}{2}\int\partial^{4}xA_{\nu}(\partial_{\mu}\partial^{\mu}A^{\nu}-\partial_{\mu}\partial^{\nu}A^{\mu})$$
$$\frac{1}{2}\int\partial^{4}xA_{\nu}(\square n_{\mu\nu}-\partial_{\mu}\partial^{\nu})A^{\mu}$$

That's pretty close. Start with the first integral and do it one step at a time.

1. First, replace $\partial_\mu\partial^\mu$ with $\Box$.
2. Then raise the index on $A_\nu$. When you do that, you have to lower $\nu$ on the other terms.
3. Next, replace $A_\nu$ (it should only appear once at this point) with $A_\nu = \eta_{\mu\nu} A^\mu$.
4. Now you should have $A^\mu$ on the right on both terms, so you can pull it out to the right.

At this point, you should have
$$\frac{1}{2}\int d^4x\,A^\nu(\Box \eta_{\mu\nu}-\partial_\mu\partial_\nu) A^\mu$$ Now if you want to, you can relabel $\mu \leftrightarrow \nu$ so that it'll match up exactly with the expression from the homework assignment.