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How would I expand the following:

$$(\partial_\mu A^\mu)^2 \tag{1}$$

My understanding of it makes me think it would be as simple as:

$$(\partial_\mu A^\mu)(\partial_\mu A^\mu)\tag{2}$$

but I recall in my lectures seeing something like:

$$\tag{3}(\partial_\mu A^\mu)^2 = (\partial_\mu A^\mu)(\partial_\nu A^\nu) $$

Which one of thee is correct? If none of them are, how would I proceed on expanding it?

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1 Answer 1

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It should be written as (3), because we only want to use a given label for one pair of contracted indices. A much simpler example is $(\sum_ia_i)^2=\sum_ia_i\cdot\sum_ja_j=\sum_{ij}a_ia_j$. (Or, if you want an example with contraction, $(\sum_ia_ib_i)^2=\sum_{ij}a_ib_ia_jb_j$.)

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  • $\begingroup$ So in $(3)$ would the differentiation in the first term affect in any way the second term? My instinct says it would not , that due to the closed brackets the terms wouldn't interact.Is this accurate? $\endgroup$ Commented May 31, 2020 at 20:05
  • $\begingroup$ @user7077252 That's right, we just multiply the sums. $\endgroup$
    – J.G.
    Commented May 31, 2020 at 20:17

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