- #1
kochanskij
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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?
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?