Integrating by parts Maxwell Lagrangian

AI Thread Summary
The discussion focuses on rewriting the free electromagnetic action integral using integration by parts. The user seeks clarification on transitioning from one line of the equation to another while managing the indices of the gauge fields correctly. They mention that surface terms can be disregarded but are unsure how to proceed with the integration process. Additionally, they inquire about any texts that might provide a derivation of this process. The conversation emphasizes the complexities of handling gauge field indices during integration by parts.
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I attached a file that shows the free EM action integral and how it can be rewritten. I would like to know how to go from the first line to the second. I have to integrate by parts somehow, and I know surface terms get thrown out, but I do not know how the indices of the gauge fields should be handled.

thanks for any help
 

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Has then perhaps anybody seen a text were it is derived?

I got -A∂∂A + A∂∂A, and both terms upper indices for A∂ and lower indices for ∂A. But what now? How to get to the last line?
 
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