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At the start of QFT, studying the Klein-Gordon scalar field, it is often mentioned that the following is the definition of the scalar product in the space of the solutions:

$$\langle f _{\vec{k}}|f_{\vec{k}'}\rangle \ \dot= \ i \int d^3x f_{\vec{k}}^*(x)\overleftrightarrow{\partial}_tf_{\vec{k}'}(x)$$

My question is: what is the precise definition of the operator $\overleftrightarrow{\partial}_t$, and what is the name of this operator? I have always seen it in formulas without any definition, or explaination, or nomenclature, and this leaves me with a feeling of uncertainty. I would like a proper nomenclature and definition for this symbol.

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This is the definition you are looking for.

$$ f\overleftrightarrow{\partial_\mu }g \equiv f\partial_\mu g-(\partial_\mu f)g$$ This is taken from Srednicki chapter 3 near eq. 3.21. To apply it to your case, just set $\mu=0$.

I don’t know if it has a special name, but I never heard of one. Relevant: The commenter notes this is not standard outside of QFT context and I concur.

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