Is the step of analytic continuation unavoidable or can you model around it?

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.

In cases where analytic continuation is applicable, does this suggest there is a formulation of the theory at hand, which directly leads to these result?

That would be a theory where the "modified" interpretation of the mathematical quantities might be taken to by 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?

Answer 1

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 – especially functions of momenta or energy – that don't admit an analytic continuation that may be classified as "unnatural", "awkward", "man-made", "contrived", if not "pathological". More generally, every proposed theory that uses non-analytic functions or doesn't allow us to continue observables to complex values is unnatural.

While some of these proclamations may look "philosophical" and "subjective" and they may lead to controversies, there's still an important general point about science. Science is about finding theories that agree with the observations. Quantum field theories and string theory do – and they often use analytic continuation in the calculations. So this is evidence that analytic continuation is helpful in Nature and in this sense, it is natural. In the previous paragraph, I said and meant that the analytic continuation was natural in a more general, less empirical sense, too.

In QFT, there are lots of calculations in which the expressions in the Euclidean spacetime – which are analytically continued – are actually more well-behaved and more well-defined than those in the Minkowski spacetime. That's just how the maths works. For example, the signature is $++++$ in the Euclidean space and the balls of fixed proper length are compact (instead of hyperboloids we know from the Minkowski signature). The Feynman path integral has $\exp(-S)$ in the Euclidean space which is more easily convergent (for action $S$ bounded from below) than the Minkowski $\exp(iS)$. In fact, the Feynman path integral may be rigorously defined in the theory of Lebesgue measure but only in the Euclidean signature.

One also uses analytic continuation to a complex dimension $d$, the so-called dimensional regularization, in QFTs. This may be viewed as a mere mathematical trick but it's a very useful one because it preserves some symmetries. For the same reason, the zeta-function regularization that allows us to calculate $1+2+3+4+5+\dots = -1/12$, is a very powerful tool in conformal field theories because it's "natural" in the sense that it doesn't invalidate any symmetries (conformal symmetry) and other underlying mathematical structures.

The analytic continuation of scattering amplitudes to complex values of momenta has poles and these poles exactly know about the location of bound states (negative interaction energy). Also, thermal density matrices are evolution operators with complex values of time. String theory's world sheet are almost always visualized with the Euclidean signature because the topology of Riemann surfaces is pretty for the Euclidean signature and it would be confusing, to say the least, if the signature were Minkowskian. The observables in the Minkowski space may always be obtained by continuing back from the Euclidean space time or world sheet to the physical Minkowski one.

So your whole question is based on an invalid understanding what is natural and what is not when it comes to the analytic continuation. The right way is to turn your expectations upside down and learn this technique that – as is shown not only by the empirical evidence – is important for a proper analysis of Nature. One may do QFT and string theory without analytic continuation, or even without complex numbers, but it would be much more messy and there would have to be many steps whose goal would be to emulate analytic continuation (and complex numbers), anyway, without using the right words.

Answer 2

Any analytic function is defined everywhere on its Riemannian surface just by its values in an arbitrarily small neighborhood of a point.

The expression to be ''analytically continued'' therefore just specifies which function is meant, but it has ''direct and natural'' values everywhere on its Riemann surface. Except that not all of these values can be calculated by the same recipe as that used for its definition. Recipes valid in diferent regions of the Riemann surface are properties of the same function complementing each other, and all of them could be used as their definition. That one defines them as an infinite sum (say) may just be a convenent starting point.

Analytic continuation is unavoidable whenever the Riemannian surface is multisheeted. This happens already for simple functions such as the square root or the log. But also for the Gamma function there is (as far as I know) no definition that covers the whole domain of the Gamma function.