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If I have some action that depends on a set of variables like so:

$S(\{\phi_{i} \}) = \phi_{i}A^{-1}_{ij}\phi_{j} + g(\phi_{i})$

(where einstein summation notation is being used for the term $\phi_{i}A^{-1}_{ij}\phi_{j}$ and $g$ is just some function)

How could I taylor expand the action around a specific value of $\phi$, say $a$, so in powers of $\phi_{i}-a$? I know I must in the end have a term like $S(a)$, and then further terms that depend on derivatives of $S$. What trips me up is that $S$ depends on $\phi_i$ and $\phi_j$, though I do see that the indexes here are not actually meaningful.

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    $\begingroup$ "discrete functional derivative" sounds just like a standard derivative (or gradient) $\endgroup$
    – fqq
    Nov 16, 2020 at 21:38
  • $\begingroup$ just to expand on the above comment, the discretization of the functional derivative $\delta/\delta \phi(x)$ would be a standard partial derivative with respect to $\phi_i$. $\endgroup$
    – Andrew
    Nov 16, 2020 at 22:25
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    $\begingroup$ Also the indices are meaningful!! $x_i$ and $x_j$ are different spacetime points if $i\neq j$. If you expand, you should take $\phi_i\rightarrow \phi_i + a_i$, in other words there should be a different variation at each spacetime point. $\endgroup$
    – Andrew
    Nov 16, 2020 at 22:30

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