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There are two different ways of dealing with infinities: There is renormalization which has nothing to do with substituting some value for the cutoff. Rather, parameters of the Lagrangian are expressed in terms of measurable quantities which effectively hides the divergences. Another way of trying to get information out of these infinities is to treat a ...


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In fact, analytical continuation in some sense is subtracting off the divergence using local counter terms. A very clear treatment has been made in Terry Tao's blog: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ (See section 2) Note the result of the ...


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Your example has no counterterms to cancel the 1/r singularity! This explains the discrepancy. In a correctly regularized expression, the counterterms are not introduced in an ad hoc way but by renormalizing some constants in the original problem that gives rise to the series. These modify all terms in the sum and make the divergent part vanish for ...


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I don't know much about QFT at all, but I have read some discussions of Casimir's effect in Matthew Schwartz's QFT textbook and I hope I can offer some valuable ideas about why the different regulators agree in some cases but not others. Essentially, in Schwartz's treatment, there is a class of regulators that yield the same physical result, namely the $-\...



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