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In renormalization of physics, $$\sum_{j=1}^{\infty}j=-\frac{1}{12}$$ We may obtain the result in two ways: first we may redifine the sum so we have used two system of math with different definition of sum, secondly, we may obtain the result by analytic continuity. Obviously, the second one is in the same system of math. So in the case, renormalization may use method in a same system to obtain the result.

But I have a question about this.

Is there any case of remormalization in which we have to solve it by way in two different systems?

I can not find a proper place or bbs to ask the question, so post it here, and if it is ambiguious, please let me know and clarify it.

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    $\begingroup$ I presume you are referring to regularization of a divergent sum, as in the Casimir effect (where the left-hand-side of the equation is the difference of sum and integral, and the right-hand-side is 1/12); I have no idea what you mean by "different system of math"; there are different methods of regularization, and if referring to the same physical effect they must produce the same answer. $\endgroup$
    – Carlo Beenakker
    Commented Oct 23, 2023 at 6:22
  • $\begingroup$ @CarloBeenakkerc， thank you for your comment, in physics of Chinese, I think, regularization and renormalization are mixed, so what I mean is regularization. Different system of math means for instance, summation has two differenct definitions, and one definition can not cover another. In other word, in fact, there are different summation with the same symbol. $\endgroup$
    – XL _At_Here_There
    Commented Oct 23, 2023 at 9:36
  • $\begingroup$ "summation has two different definitions" --- I really have no idea what that means. $\endgroup$
    – Carlo Beenakker
    Commented Oct 23, 2023 at 10:48
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    $\begingroup$ I think "summation has two different definitions" is referring to divergent series where different summation methods give different results. Such examples appear in Summation methods for divergent series. Now the question seems to be asking whether such examples occur in the context of regularization in QFT. $\endgroup$
    – Tobias Fritz
    Commented Oct 23, 2023 at 13:08
  • $\begingroup$ @TobiasFritz, Yes, that is what I mean $\endgroup$
    – XL _At_Here_There
    Commented Oct 23, 2023 at 13:49

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A charitable reading suggests that you are referring to the $\zeta$ function regularization $$ \sum_{i=1}^\infty i = \lim_{s\to -1} \sum_{i=1}^\infty i^{-s}=\zeta(-1)=-\frac{1}{12} $$ which occurs in calculations of the Casimir effect, and in a derivation of the Stefan-Boltzmann law in thermal field theory.

While regularization is a necessary prerequisite to renormalization, it is by no means the same thing, so the question as posed is unclear at best. Moreover, it is by no means clear what you mean by "different systems of math". If you are referring to different ways to assign values to divergent sums, then I'm not aware of any situation where something other than $\zeta$ function regularization is used on mode sums (this is largely due to the fact that it is quite closely related to dimensional regularization).

On the other hand, different forms of regularization such as Borel resummation are frequently used in the context of dealing with divergent series in the perturbative coupling. Again, the choice of method is not entirely arbitrary, but has to do with the expected growth behavior of the coefficients in such series. So, I would tend to think that the answer to your question is "no", i.e. there is no context where the comparison of two different methods of regularizing divergent series is required.

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