1+2+3+4+ = -1/12

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Hello everybody!

I'm actually a mathematician but occasionally I read stuff on quantum physics for fun and a few days ago I read something that really surprised me. Since then I was trying to get an idea about the actual mathematics behind my discovery but couldn't. So here I am hoping that some quantum physics expert can show me the actual calculations or at least point me in the right direction! About my question:

Being a mathematician I know that there are several good reasons to claim that the value of the effectively divergent series 1+2+3+4+... (i.e. the sum of all natural numbers) might in fact be -1/12 (http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯). The modern argument is as follows: If Zeta is the Riemann Zeta function (defined for all complex numbers but 1 by analytic continuation) then Zeta(-1) = -1/12. But Zeta(-1) also corresponds to the divergent series 1+2+3+4+... as Zeta(s) is given by the infinite sum 1/n^s, where n goes from 1 to infinity, if the real part of s is greater than 1. Using this definition Zeta(-1) would be (if it was defined there) 1/1^(-1)+ 1/2^(-1)+ 1/3^(-1)+ 1/4^(-1)+... = 1+2+3+4+... . So Zeta(-1) is "the sum of all natural numbers" on the one hand and -1/12 on the other hand which is why mathematicians, if forced to assign a value to the sum of all natural numbers, prefer -1/12 over all other possible values. From a mathematicians point of few this is not very surprising. It's the same thing as claiming that 5.5 factorial (which is undefined) is actually Gamma(6.5) or that the sum of all powers of 2 is -1 (the geometric series sum x^n where n goes from 0 to infinity is only defined if -1<x<1 but it has the analytic continuation 1/(1-x) which is -1 for x=2). I never thought that this result, 1+2+3+4+... = -1/12, could ever have any physics application but I read at many places that it is actually used in string theory! One application is claimed to be the computation of the Casimir effect which pushes two plates in the vacuum together. While outside the two plates virtual particles of all wavelengths can occur, in between the two plates only certain wavelengths are allowed. This pushes the plates together and to get the actual strength of the resulting force one has to sum over all wavelengths. At this point the sum of all natural numbers allegedly comes in. If you replace it by -1/12 you get the correct result, i.e. what is observed in experiments. The minus even gives you the direction of the force (pushing, not pulling). Is there anyone who actually understands the computations behind this and can show me where the substitution 1+2+3+4+... = -1/12 comes in? This is very intriguing to me and I would appreciate any help to understand it better!
 

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Orodruin
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Great, I will have a look! Seems to be exactly what I was looking for, thanks a lot!
 

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