# How to perform discrete functional derivative

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.

• "discrete functional derivative" sounds just like a standard derivative (or gradient)
– fqq
Commented Nov 16, 2020 at 21:38
• 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 $\phi_i$. Commented Nov 16, 2020 at 22:25
• 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. Commented Nov 16, 2020 at 22:30