What physicists mean when they say 1+2+3+... = -1/12pic.twitter.com/Noj5pEf1Ym
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I want to be clear. I do not assert any "equality" of the sum to -1/12 in the limit that ε-->0. This makes no sense What does make sense is to speak about a "finite piece" and a divergent piece. This result is unique, as outlined by Terrance Tao.https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ …
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Sometimes in physics, absolute quantities don't have a meaning. Rather what does have a meaning is *differences* . In these situations, the term that *would* diverge cancels out completely leaving behind a finite answer; giving way to the following "replacement rule"pic.twitter.com/QHm3LbXoai
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I find this really not convincing. It feels like you just saying S= -1/12 + \infty. But for any constant A you could write S=A+\infty so I don’t see what’s special about -1/12. I find the analytic continuation explanation much more convincing. Unless I’m missing something here?
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The limit is never actually taken to zero. The 1/ε^2 actually tells us precisely about the pole structure of the function. There is no assertion about an "equality" when ε-> 0. It does however, make sense to speak about a "convergent piece" and divergent piece in this limit.
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When a friend showed me a similar derivation in grad school, I noticed the appearance of the Koebe function z/(1-z)^2 there, too. (See https://cornellmath.wordpress.com/2007/07/30/sum-divergent-series-ii/ ….) Any reason Schlicht functions should be connected with all this?
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I'm not too familiar with Schlicht functions but it looks like it's just an analytic that satisfies certain properties which aren't related to this in an obvious way, I think
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This is fascinating. Is there a physical application that causes physicists to cite this?
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Indeed! the classic example is in the casimir effect! I'll try to tweet about that soon
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Shouldn't (a) start at n=0 for the geometric series or subtract a 1 from the RHS? It doesn't matter as the derivative of a constant would lead to the same (b), but (a) has an error.
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Yes indeed it should! The typo doesn't effect the result though luckily since the derivative kills the first term
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