What physicists mean when they say 1+2+3+... = -1/12pic.twitter.com/Noj5pEf1Ym
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When we analytically continue the series for the riemann zeta function to this value, we get -1/12 without the divergence, right? Why does the analytic continuation act like some kind of MS scheme? I've never noticed.
You're right.. it's an amazing fact. A priori, there's no reason why if we naively use the series rep of the ζ function outside of its domain of validity (Re(s)>1) that we should get "ζ(-1) + divergent piece"..
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I came here for these comments. I'd love to know if anyone has more insight. At first sight I was like "you mean mathematician" then I read the slide. Nope, you do mean physicist... I never knew this! Can other "renormalisation" tricks be viewed somehow as analytic continuation?
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Well... that's the thing. The majority of renormalization in quantum field theories is in practise done by formally analytically continuing divergent quantities in the number of dimensions, which is taken to be d \in C. The divergences 1/\epsilon can then be dealt with. But 1/2
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