How Do You Prove Electromagnetic Field Quantization?

[smallgirl]

This:

 

[vela]

That doesn't really explain what you're thinking. What you're doing is like taking x(ay+bx) and saying it's equal to x(a+b)y. You're just replacing one variable by another for no apparent reason.

 

[smallgirl]

Hmm... I figured I could rearrange and use the results of integration by parts to achieve the rearrangement that I did.. But I guess what I did was wrong...

 

[smallgirl]

I'm thinking I might need to permute the indices and do some more raising and lowering?

$$\frac{1}{2}\int\partial^{4}x(-A_{v}\partial_{\mu}\partial^{\nu}A^{\mu}+A_{\nu}\partial_{\mu}\partial^{\mu}A^{\nu})$$


$$\frac{1}{2}\int\partial^{4}x(A_{v}\partial_{\mu}\partial^{\mu}A^{\nu}-A_{\nu}\partial_{\mu}\partial^{\nu}A^{\mu})$$


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


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

 

[vela]

Do you understand why what you did was wrong? Do you see that $A_\nu \partial_\mu \partial^\mu A^\nu$ and $A_\nu \partial_\mu \partial^\mu A^\mu$ aren't equal to each other? You've now made this same mistake several times, so it seems like you're just pushing symbols around without any understanding.

 

[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.