One sometimes considers the analytic continuation of certain quantities in physi

Question

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually.

For example if you use the procedure for regularization, it sometimes seems like an ad hoc step.

Question: In cases where analytic continuation is applicable, does this suggest there is another formulation of that theory, which leads to these result directly?

That would be a theory where the "modified" interpretation of the mathematical quantities might be taken to be a starting point.

For example if you define some fundamental quantity of your theory as an integral or sum, and it doesn't converge somewhere, and you make an analytic continuation to get some valuable results. Could this imply there is a formulation where that value comes naturally, i.e. a formulation where there never is this sum object which makes problems?

Explanation / Answer

Could this imply there is a formulation where that value comes naturally...

This sentence implicitly assumes that analytic continuation is "unnatural". But the truth is the other way around: analytic continuation is one of the most natural mathematical procedures in physics. On the contrary, it's functions

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