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Analytic continuation and physics

  1. May 10, 2015 #1
    Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite series that occurs in a physical theory of nature? Will this process give a correct answer that can be verified by a physics lab experiment?

    A specific example:

    The infinite series 1+2+3+4+5+.... occurs in a calculation for the Casimir Effect. This series is the Zeta Function at s = -1. Z(s) = 1/1^s + 1/2^s + 1/3^s + .... The analytic continuation of Z(-1) is -1/12 Is it correct to say that 1+2+3+4+5+.... = -1/12 ? If you use the value -1/12 in your calculation of the Casimir force acting on parallel conducting plates and then measure this force in the lab, will your calculation agree with experiment?

    Another example:

    The Zeta function at s = 0 yields the series 1+1+1+1+1+.... The analytic continuation of Z(0) is -1/2 This series occurs in String Theory when calculating the number of space-time dimensions. Is it correct to say that 1+1+1+1+1+.... = -1/2 and use this value in your calculation? The result comes out to be 10 dimensions. Should we expect nature to agree with this calculation?
  2. jcsd
  3. May 10, 2015 #2


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    If the physical theory (like in the cases you described) can be modeled with the Zeta function, then it's OK.
  4. May 13, 2015 #3
    In the physical theories that I refer to, the Zeta function does NOT occur. Just the infinite series 1+2+3+4+.... or the infinite series 1+1+1+1+.... occurs. My question is: Is it correct to equate the finite values of -1/12 and -1/2 to these series?
    It seems absurd to claim that the sum of an infinite amount of positive integers can equal a negative fraction! Can anyone justify this claim? (I understand analytic continuation, but I still don't think it applies to these diverging series.

    A contradiction: In the first series, replace 2 with 1+1, 3 with 1+1+1, 4 with 1+1+1+1, and so on. We then have the sum of an infinite number of 1's. So these two series should sum to the same value. But analytic continuation gives -1/12 for the first series and -1/2 for the second. Can this contradiction be explained?
  5. May 18, 2015 #4


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    The formula ## \zeta(0)=-\frac{1}{2}## is proved here


    using the functional equation for ## \zeta ## and the formula for ## \zeta ## in the critical strip, the second ##1+ 2+3+4+ ... =-\frac{1}{12}## is the famous Ramanujan sum, you can see a proof here


    the criterion to obtain these kinds of series is different from the classic criterion of convergence , this is a sort of ''mean convergence'' (it is not a contradiction, it is another way to interpret a sum ...), as for exampe

    [tex] 1-1+1-1+....=\frac{1}{2}[/tex]

    if you do the average of the partial sums you have ## \frac{1+0}{2}=\frac{1}{2}, \frac{1+0+1}{3}=\frac{2}{3}, \frac{1+0+1+0}{4}=\frac{1}{2}, ... ## and these converge to ## \frac{1}{2}##

    you can use the analytic continuation to extend ##\zeta## to this value.

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